Recent zbMATH articles in MSC 53Chttps://zbmath.org/atom/cc/53C2021-05-28T16:06:00+00:00WerkzeugCR-submanifolds of Chen-type two in non-flat complex space forms.https://zbmath.org/1459.530572021-05-28T16:06:00+00:00"Dimitrić, Ivko"https://zbmath.org/authors/?q=ai:dimitric.ivko"Djorić, Mirjana"https://zbmath.org/authors/?q=ai:djoric.mirjanaSummary: We prove some nonexistence results for certain families of CR-submanifolds of Chen-type two in complex space forms. For example, there exist no holomorphic submanifolds of the complex hyperbolic space which are of 2-type via the standard embedding by projectors. This is in contrast to the situation in complex projective space, which is known to contain Einstein-Kähler submanifolds of 2-type. We further show that there are no mass-symmetric ruled real hypersurfaces in \(\mathbb{C}Q^m(4c)\) of Chen-type 2. Additionally, we characterize mass-symmetric totally real submanifolds of 2-type in terms of detailed intrinsic and extrinsic conditions and derive some corollaries for Lagrangian submanifolds.
For the entire collection see [Zbl 1454.53006].Maximally-warped metrics with harmonic curvature.https://zbmath.org/1459.530452021-05-28T16:06:00+00:00"Derdzinski, Andrzej"https://zbmath.org/authors/?q=ai:derdzinski.andrzej"Piccione, Paolo"https://zbmath.org/authors/?q=ai:piccione.paoloSummary: We describe the local structure of Riemannian manifolds with harmonic curvature which admit a maximum number, in a well-defined sense, of local warped-product decompositions, and at the same time their Ricci tensor has, at some point, only simple eigenvalues. We also prove that, in every given dimension greater than two, the local-isometry types of such manifolds form a finite-dimensional moduli space, and a nonempty open subset of this moduli space is realized by locally irreducible complete metrics which are neither Ricci-parallel, nor -- for dimensions greater than three -- conformally flat.
For the entire collection see [Zbl 1454.53006].On stability and index of minimal submanifolds.https://zbmath.org/1459.530612021-05-28T16:06:00+00:00"Chen, Hang"https://zbmath.org/authors/?q=ai:chen.hang|chen.hang.1Summary: Minimal submanifolds are critical points of the volume functional. It plays an important role in the geometry of submanifolds and has been widely and deeply studied. In this paper, we survey the stability and the index of minimal submanifolds in several special ambient spaces. In the first part, we mainly summarize known results on the existence and the classification of closed stable submanifolds of arbitrary codimension in compact rank one symmetric spaces and product manifolds. In the second part, we exhibit results on minimal hypersurfaces with lower index in a sphere, a projective space and the product manifold of two spheres, including a new result of the author.
For the entire collection see [Zbl 1454.53006].Reflections on some research work of Bang-Yen Chen.https://zbmath.org/1459.530022021-05-28T16:06:00+00:00"Van der Veken, Joeri"https://zbmath.org/authors/?q=ai:van-der-veken.joeri"Carriazo, Alfonso"https://zbmath.org/authors/?q=ai:carriazo.alfonso"Dimitrić, Ivko"https://zbmath.org/authors/?q=ai:dimitric.ivko"Oh, Yun Myung"https://zbmath.org/authors/?q=ai:oh.yun-myung"Suceavă, Bogdan D."https://zbmath.org/authors/?q=ai:suceava.bogdan-dragos"Vrancken, Luc"https://zbmath.org/authors/?q=ai:vrancken.lucSummary: This essay provides a brief sketch of selected mathematical research work of Bang-Yen Chen done during the last fifty years.
For the entire collection see [Zbl 1454.53006].Quasi contact metric manifolds with Killing characteristic vector fields.https://zbmath.org/1459.530402021-05-28T16:06:00+00:00"Bae, Jihong"https://zbmath.org/authors/?q=ai:bae.jihong"Jang, Yeongjae"https://zbmath.org/authors/?q=ai:jang.yeongjae"Park, JeongHyeong"https://zbmath.org/authors/?q=ai:park.jeonghyeong"Sekigawa, Kouei"https://zbmath.org/authors/?q=ai:sekigawa.koueiSummary: An almost contact metric manifold is called a quasi contact metric manifold if the corresponding almost Hermitian cone is a quasi Kähler manifold, which was introduced by \textit{Y. Tashiro} [Tohoku Math. J. (2) 15, 167--175 (1963; Zbl 0126.38003)] as a contact \(O^*\)-manifold. In this paper, we show that a quasi contact metric manifold with Killing characteristic vector field is a \(K\)-contact manifold. This provides an extension of the definition of \(K\)-contact manifold.On generalized \((\kappa,\mu)\)-space forms and their invariant submanifolds with quarter symmetric metric connections.https://zbmath.org/1459.530252021-05-28T16:06:00+00:00"Biswas, Nirmal"https://zbmath.org/authors/?q=ai:biswas.nirmalSummary: In this paper we find some relations between curvature tensors with respect to quarter symmetric and Levi-Civita connections. We study the second order parallel tensor in a generalized \((\kappa,\mu)\)-space form with respect to the quarter symmetric metric connection. We also study invariant submanifold of a generalized \((\kappa,\mu)\)-space form with respect to quarter symmetric metric connection.On conformal Gauss maps.https://zbmath.org/1459.530172021-05-28T16:06:00+00:00"Burstall, F. E."https://zbmath.org/authors/?q=ai:burstall.francis-eSummary: We characterise the maps into the space of 2-spheres in \(S^n\) that are the conformal Gauss maps of conformal immersions of a surface into \(S^n\). In particular, we give an invariant formulation and efficient proof of a characterisation, due to \textit{J. F. Dorfmeister} and \textit{P. Wang} [``Willmore surfaces in spheres via loop groups. I: Generic cases and some examples'', Preprint, \url{arXiv:1301.2756}], of the harmonic maps that are conformal Gauss maps of Willmore surfaces.Feeling the heat in a group of Heisenberg type.https://zbmath.org/1459.352262021-05-28T16:06:00+00:00"Garofalo, Nicola"https://zbmath.org/authors/?q=ai:garofalo.nicola"Tralli, Giulio"https://zbmath.org/authors/?q=ai:tralli.giulioSummary: In this paper we use the heat equation in a group of Heisenberg type \(\mathbb{G}\) to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators \(\mathscr{L}^s\) and \(\mathscr{L}_s\), \(0 < s \leq 1\). Here, \(\mathcal{L}^s\) is the fractional power of the horizontal Laplacian, and \(\mathscr{L}_s\) is the conformal fractional power of the horizontal Laplacian on \(\mathbb{G}\). One of our main objective is compute explicitly the fundamental solutions of these nonlocal operators by a new approach exclusively based on partial differential equations and semigroup methods. When \(s = 1\) our results recapture the famous fundamental solution found by Folland and generalised by Kaplan.Generalizations of Bonnet-Myers theorem on Finsler manifolds.https://zbmath.org/1459.530722021-05-28T16:06:00+00:00"Shen, Bin"https://zbmath.org/authors/?q=ai:shen.bin"Zhao, Zisu"https://zbmath.org/authors/?q=ai:zhao.zisuSummary: In this note, we generalize the Myers theorem on Finsler manifolds with four different curvature conditions. The first one is the ordinary Ricci curvature on Finsler manifolds. The second one is the mean Ricci curvature defined by the first author in [Kodai Math. J. 41, No. 1, 1--15 (2018; Zbl 1414.53016)]. The last two ones are the weighted Ricci curvatures which also play important roles in weighted Riemannian spaces. We also provide the generalized Myers theorem on weighted Riemannian manifolds.Pyramid Ricci flow in higher dimensions.https://zbmath.org/1459.530862021-05-28T16:06:00+00:00"McLeod, Andrew D."https://zbmath.org/authors/?q=ai:mcleod.andrew-d"Topping, Peter M."https://zbmath.org/authors/?q=ai:topping.peter-milesThis paper is devoted to the study of pyramid Ricci flow from an \(\text{IC}_1\)-limit space. The authors use a constructive method. They construct a pyramid Ricci flow starting with the noncollapsed \(\text{IC}_1\)-limit space. These spaces are global homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder. The definition and basic properties of PIC\(_1\) are given. Here, PIC\(_1\) denotes an interesting variant of positive isotropic curvature. IC\(_1\)-limit spaces are globally smooth manifolds. The presentation of this content is clear and in detailed manner. The results from this work represent an advanced study in these areas.
Reviewer: Costache Apreutesei (Iaşi)Relativity and singularities: a brief introduction for mathematicians.https://zbmath.org/1459.830102021-05-28T16:06:00+00:00"Natário, José"https://zbmath.org/authors/?q=ai:natario.joseSummary: We summarize the main ideas of General Relativity and Lorentzian Geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose Singularity Theorems. The reader is assumed to be familiar with Riemannian Geometry and point set Topology.Ringel duality for perverse sheaves on hypertoric varieties.https://zbmath.org/1459.140052021-05-28T16:06:00+00:00"Braden, Tom"https://zbmath.org/authors/?q=ai:braden.tom"Mautner, Carl"https://zbmath.org/authors/?q=ai:mautner.carlSummary: Motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities, we study a category of perverse sheaves with coefficients in a field \(k\) on any affine unimodular hypertoric variety \(\mathfrak{M}\). Our main result is that this is a highest weight category whose Ringel dual is the corresponding category for the Gale dual hypertoric variety \(\mathfrak{M}^!\). On the way to proving our main result, we confirm a conjecture of \textit{M. V. Finkelberg} and \textit{D. V. Kubrak} [Funct. Anal. Appl. 49, No. 2, 135--141 (2015; Zbl 1348.14044); translation from Funkts. Anal. Prilozh. 49, No. 2, 70--78 (2015)] in the case of hypertoric varieties. We also show that our category is equivalent to representations of a combinatorially-defined algebra, recently introduced in a related paper.An Alexandrov theorem in Minkowski spacetime.https://zbmath.org/1459.530542021-05-28T16:06:00+00:00"Hijazi, Oussama"https://zbmath.org/authors/?q=ai:hijazi.oussama"Montiel, Sebastián"https://zbmath.org/authors/?q=ai:montiel.sebastian"Raulot, Simon"https://zbmath.org/authors/?q=ai:raulot.simonA codimension-two submanifold \(\Sigma\) of a Lorentzian manifold is said to have constant normalized null curvature (CNNC) if there exists a future null normal vector field \(\mathcal L\) such that \(\Sigma\) is torsion-free with respect to \(L\) and \(\langle\mathcal H, \mathcal L\rangle\) is a constant, where \(\mathcal H\) denotes the mean curvature vector field on \(\Sigma^n\).
In the paper, using a spinorial approach developed by the first two authors and \textit{X. Zhang}, the authors generalize a theorem à la Alexandrov of [J. Differ. Geom. 105, No. 2, 249--290 (2017; Zbl 1380.53089)] to closed codimension-two space-like submanifolds in the Minkowski spacetime for an adapted CMC condition. In particular, they prove that if \(\Sigma\) is an untrapped codimension-two submanifold in the Minkowski spacetime and suppose that \(\Sigma\) has CNNC with respect to a future null normal vector field \(\mathcal L\), then \(\Sigma\) lies in a shearfree null hypersurface.
Reviewer: Anna Fino (Torino)A large family of projectively equivalent \(C^0\)-Finsler manifolds.https://zbmath.org/1459.530702021-05-28T16:06:00+00:00"Fukuoka, Ryuichi"https://zbmath.org/authors/?q=ai:fukuoka.ryuichiSummary: A \(C^0\)-Finsler structure on a differentiable manifold is a continuous real valued function defined on its tangent bundle such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent \(C^0\)-Finsler manifolds \((\hat M,\hat F)\), where \(\hat M\) is diffeomorphic to the Euclidean plane. The structures \(\hat F\) don't have partial derivatives and they aren't invariant by any transformation group of \(\hat M\). For every \(p,q \in (\hat M,\hat F)\), we determine the unique minimizing path connecting \(p\) and \(q\). They are line segments parallel to the vectors \((\sqrt{3}/2,1/2), (0,1)\) or \((-\sqrt{3}/2,1/2)\), or else a concatenation of two of these line segments. Moreover \((\hat M,\hat F)\) aren't Busemann \(G\)-spaces and they don't admit any bounded open \(\hat F\)-strongly convex subsets. Other geodesic properties of \((\hat M,\hat F)\) are also studied.Nonuniqueness for a fully nonlinear, degenerate elliptic boundary-value problem in conformal geometry.https://zbmath.org/1459.351672021-05-28T16:06:00+00:00"Shan, Zhengyang"https://zbmath.org/authors/?q=ai:shan.zhengyangSummary: One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing \(\sigma_k\)-curvature in the interior and constant \(H_k\)-curvature on the boundary. When restricting to the closure of the positive \(k\)-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove nonuniqueness for the boundary-value problem \(\sigma_4\)-curvature equals zero and constant \(H_4\)-curvature by using bifurcation results proven by Case, Moreira and Wang. Surprisingly, our construction via products of sphere and hyperbolic space only works for a finite set of dimensions.A finite quotient of join in Alexandrov geometry.https://zbmath.org/1459.530482021-05-28T16:06:00+00:00"Rong, Xiaochun"https://zbmath.org/authors/?q=ai:rong.xiaochun"Wang, Yusheng"https://zbmath.org/authors/?q=ai:wang.yushengAlexandrov geometry was introduced by \textit{Yu. Burago} et al. [Russ. Math. Surv. 47, No. 2, 1 (1992; Zbl 0802.53018); translation from Usp. Mat. Nauk 47, No. 2(284), 3--51 (1992)], and extensive study has been done since then. An Alexandrov space with curvature greater than or equal to \(k\) is a locally complete length metric space such that every geodesic triangle looks fatter than a corresponding triangle in the simply connected 2-space form of constant curvature greater than or equal to \(k\). Several authors studied the Alexandrov geometry [\textit{T. Frankel}, Pacific J. Math. 11, 165--174 (1961; Zbl 0107.39002); \textit{S. Y. Cheng}, Math. Z. 143, No. 3, 289--297 (1975; Zbl 0329.53035); \textit{D. Gromoll} and \textit{K. Grove}, Ann. Sci. École Norm. Sup. (4) 20, No. 2, 227--239 (1987; Zbl 0626.53032); \textit{K. Grove} and \textit{S. Markvorsen}, J. Am. Math. Soc. 8, No. 1, 1--28 (1995; Zbl 0829.53033); \textit{K. Grove} and \textit{K. Shiohama}, Ann. of Math. (2) 106, No. 2, 201--211 (1977; Zbl 0341.53029); \textit{G. Perelman}, J. Differ. Geom. 40, No. 1, 209--212 (1994; Zbl 0818.53056); \textit{A. Petrunin}, Geom. Funct. Anal. 8, No. 1, 123--148 (1998; Zbl 0903.53045)].
The principal objective in this paper is to explore a rigidity of a finite quotient of join in Alexandrov geometry, which is a necessary step toward a classification for Alexandrov spaces of curvature greater than or equal to 1 and diameter \(\frac{\pi}{2}\).
Reviewer: Lakehal Belarbi (Mostaganem)On the linear stability of nearly Kähler 6-manifolds.https://zbmath.org/1459.530532021-05-28T16:06:00+00:00"Semmelmann, Uwe"https://zbmath.org/authors/?q=ai:semmelmann.uwe"Wang, Changliang"https://zbmath.org/authors/?q=ai:wang.changliang"Wang, M. Y.-K."https://zbmath.org/authors/?q=ai:wang.mckenzie-y-kA nearly Kähler manifold \((M,J,g)\) is an almost Hermitian manifold that satisfies \((\nabla_{X} J) X=0\), for all tangent vectors \(X\), where \(\nabla\) denotes the Levi-Civita connection of \(g\). The nearly Kähler structure is strict if it is not Kähler. A closed Einstein manifold \((M, g)\) is linearly stable if for all transverse traceless (TT) symmetric \(2\)-tensors \(h\), i.e., divergence-free and trace-free
symmetric \(2\)-tensors, the quadratic form \(\mathcal{Q}(h,h)=-<\nabla^{\ast}\nabla h-2\dot{R}h,h>_{L^{2}(M,g)}\) is \(\leq 0\), where \(\dot{R}\) is the action of the curvature tensor on symmetric \(2\)-tensors. \((M, g)\) is linearly unstable if it is not linearly stable. Several authors studied stability and instability manifolds [\textit{X. Dai} et al., Invent. Math. 161, No. 1, 151--176 (2005; Zbl 1075.53042); \textit{L. Foscolo}, J. Lond. Math. Soc., II. Ser. 95, No. 2, 586--612 (2017; Zbl 1376.53067); \textit{L. Foscolo} and \textit{M. Haskins}, Ann. Math. (2) 185, No. 1, 59--130 (2017; Zbl 1381.53086); \textit{Th. Friedrich}, Math. Nachr. 97, 117--146 (1980; Zbl 0462.53027); \textit{R. Grunewald}, Ann. Global Anal. Geom. 8, No. 1, 43--59 (1990; Zbl 0704.53050); \textit{S. J. Hall} and \textit{T. Murphy}, Proc. Am. Math. Soc. 139, No. 9, 3327--3337 (2011; Zbl 1231.53056); \textit{K. Kröncke}, Calc. Var. Partial Differ. Equ. 53, No. 1--2, 265--287 (2015; Zbl 1317.53086)].
The principal objective in this paper is: Let \((M, J , g)\) be a complete strict nearly Kähler \(6\)-manifold. If \(b_{2}(M)\) or \(b_{3}(M)\) is nonzero, then \(g\) is linearly unstable with respect to the Einstein-Hilbert action restricted to the space of Riemannian metrics with constant scalar curvature and fixed volume. Hence, it is also linearly unstable with respect to the \(\nu\)-entropy of \textit{G. Perelman} [arXiv e-print service 2002, Paper No. 0211159, 39 p. (2002; Zbl 1130.53001)], and dynamically unstable
with respect to the Ricci flow.
Reviewer: Lakehal Belarbi (Mostaganem)The complete classification of a class of new linear Weingarten surfaces in Minkowski space.https://zbmath.org/1459.530322021-05-28T16:06:00+00:00"Yang, Dan"https://zbmath.org/authors/?q=ai:yang.dan"Zhu, Xiao Ying"https://zbmath.org/authors/?q=ai:zhu.xiaoyingSummary: We consider spacelike and timelike surfaces satisfying an interesting geometric property that the gradient of the mean curvature is a principal direction in a Minkowski 3-space. It is proved that these surfaces are linear Weingarten surfaces and contain all biconservative surfaces in a Minkowski 3-space. We call them \textit{generalized biconservative surfaces} (or \textit{GB surfaces} for short). We give complete explicit classifications of spacelike and timelike GB surfaces in Minkowski 3-space. Our results show that the non-constant-mean-curvature GB surfaces in Minkowski 3-space are locally either surfaces of revolution or null scrolls.Holonomy groups of compact flat solvmanifolds.https://zbmath.org/1459.220022021-05-28T16:06:00+00:00"Tolcachier, A."https://zbmath.org/authors/?q=ai:tolcachier.aA solvmanifold is a compact homogeneous space \(G/\Gamma\) of a simply connected solvable Lie group \(G\) by a discrete subgroup \(\Gamma\). In this article the author studies the holonomy groups of solvmanifolds admitting a flat Riemannian metric induced by a flat left-invariant metric on the corresponding Lie group.
\textit{J. W. Milnor} [Adv. Math. 21, 293--329 (1976; Zbl 0341.53030)] obtained a characterization of Lie groups which admit a flat left-invariant metric and he proved that they are all solvable of a very restricted form, showing that its Lie algebra decomposes orthogonally as an abelian subalgebra and an abelian ideal, where the action of the subalgebra on the ideal is by skew-adjoint endomorphisms.
By [\textit{L. Auslander} and \textit{M. Auslander}, Proc. Am. Math. Soc. 9, 933--941 (1959; Zbl 0099.39003)] the holonomy group of a flat solvmanifold is abelian. In this paper the author gives an elementary proof of this property and moreover he shows that the holonomy group of a flat almost abelian solvmanifold is cyclic.
Due to a result of [\textit{L. Auslander} and \textit{M. Kuranishi}, Ann. Math. (2) 65, 411--415 (1957; Zbl 0079.38304)], every finite group is the holonomy group of a compact flat manifold, so it is interesting to know, given a finite group \(H\), the minimal dimension of a compact flat manifold with holonomy group \(H\). Concerning this the author shows that the minimal dimension of a flat solvmanifold with holonomy group \(\mathbb Z_n\) coincides with the minimal dimension of a compact flat manifold with holonomy group \(\mathbb Z_n\), for \(n \geq 3\). Finally, he describes the possible holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6. Up to dimension 5 all flat solvmanifolds are quotients of almost abelian Lie groups, therefore their holonomy groups are cyclic. In dimension 6 he constructs examples with non-cyclic abelian holonomy groups as particular cases of a general construction.
Reviewer: Anna Fino (Torino)Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures.https://zbmath.org/1459.530492021-05-28T16:06:00+00:00"Matveev, Vladimir Sergeevich"https://zbmath.org/authors/?q=ai:matveev.vladimir-sSummary: We generalize the result of \textit{V. S. Matveev} and \textit{P. J. Topalov} [Math. Z. 238, No. 4, 833--866 (2001; Zbl 0998.53025)] to all signatures.$\eta$ invariants under degeneration to cone-edge singularities.https://zbmath.org/1459.580092021-05-28T16:06:00+00:00"Fornasin, Nelvis"https://zbmath.org/authors/?q=ai:fornasin.nelvisThe author studies the behavior of the eta-invariant of the Spin Dirac operator
when the underlying Riemannian metric degenerates to a conical metric at some isolated points.
Let us describe this degeneration more precisely for a unique singular point. Take \(\Omega_0\) to be a compact metric space isometric to a smooth Riemannian manifold in the complement of a point \(p\). Assume that the metric \(g_0\)
is conical
near \(p\) with link \(Y\), in the sense that near \(p\) it takes the form \(d\rho^2+\rho^2 h\), where \(\rho<\rho_0\) is the distance function
to \(p\), and \(h\) is a Riemannian metric on the compact manifold \(Y\). Consider also a large conical metric, i.e., a
complete manifold \((Z,g_Z)\) which outside a compact \(K\)
is isometric to the product space \(([r_0,\infty)\times Y, dr^2+r^2h)\) for the same manifold \((Y,h)\)
as above. This metric is isometric to \(g_0\) for \(\rho\in(r_0,\rho_0)\). It is also invariant by rescaling outside a compact set, so
\(\epsilon^2 g_Z\) is isometric to \(g_0\) for \(\rho\in(\epsilon r_0,\rho_0)\). The conical limit considered in this paper
means the family of compact Riemannian manifolds \(\Omega_\epsilon\) obtaind by glueing \((\Omega_0,g_0)\) and
\((Z,\epsilon^2 g_Z)\) along the region where \(\epsilon r_0<\rho<\rho_0\).
They converge to \(\Omega_0\) in the Hausdorff topology.
The definition can be easily extended to the case of several singular points.
The manifolds \(\Omega_\epsilon\) are all diffeomorphic and assumed to be endowed with a fixed spin structure.
The eta invariant of a spin manifold \(\Omega\) is defined in terms of the spectrum of the Dirac operator \(D\).
First, one considers the eta function \(\eta(D,s)=\sum_\lambda \lambda|\lambda|^{-s-1}\) where \(\lambda\)
spans the non-zero eigenvalues of \(D\) counted with their multiplicity.
This is a holomorphic function on the half-plane \(\Re(s)>\dim(\Omega)\), extends meromorphically to the complex plane, and is regular
at \(s=0\). The eta invariant is this regular value at \(s=0\). A similar definition applies to the odd signature operator.
The author proves that the limit as \(\epsilon \to 0\) of \(\eta(\Omega_\epsilon)\) can be computed in terms of the eta invariant
of the conical manifold \(\Omega_0\) (its spectrum is discrete by the work of Chow),
of a regularized eta invariant for the non-compact manifold \(Z\), and the sum of the signs of a certain
finite family of eigenvalues converging to \(0\).
More generally, the same problem is studied when the degeneration produces in the limit a cone-edge manifold,
described briefly as being
locally modeled over a Riemannian product \(\Omega_\epsilon\times B\) for some fixed compact Riemannian manifold \(B\).
As an application, the author examines the so-called extended \(\nu\)-invariant of Joyce manifolds
(torsion free spin manifolds in dimension \(7\) with holonomy group \(G_2\)). This extended \(\nu\)-invariant, a linear combination
of the eta invariants for the Dirac and the odd signature oiperators, was introduced by
Crowley, Goette and Nordström, who showed that it takes integer values on \(G_2\) manifolds. Here the extended \(\nu\)-invariants are compared for a certain cone-edge degeneration of \(G_2\) metrics.
Reviewer: Sergiu Moroianu (Bucureşti)Almost bi-slant submanifolds of an almost contact metric manifold.https://zbmath.org/1459.530432021-05-28T16:06:00+00:00"Perktaş, Selcen Yüksel"https://zbmath.org/authors/?q=ai:perktas.selcen-yuksel"Blaga, Adara M."https://zbmath.org/authors/?q=ai:blaga.adara-monica"Kılıç, Erol"https://zbmath.org/authors/?q=ai:kilic.erolSummary: In this paper we introduce and study the almost bi-slant submanifolds of an almost contact metric manifold. We give some characterization theorems for almost bi-slant submanifolds. Moreover, we obtain integrability conditions of the distributions which are involved in the definition of almost bi-slant submanifolds. We also get some results for totally geodesic and totally umbilical almost bi-slant submanifolds of cosymplectic manifolds and Sasakian manifolds.Lorentzian symmetric spaces which are Einstein-Yang-Mills with respect to invariant metric connections.https://zbmath.org/1459.530552021-05-28T16:06:00+00:00"Castrillón López, Marco"https://zbmath.org/authors/?q=ai:castrillon-lopez.marco"Gadea, P. M."https://zbmath.org/authors/?q=ai:gadea.pedro-m"Rosado María, María Eugenia"https://zbmath.org/authors/?q=ai:rosado-maria.eugeniaLorentzian symmetric Lie groups which provide solutions of Einstein-Yang-Mills equations are known from the literature. Komrakov's list contains 4-dimensional pseudo-Riemannian homogeneous spaces (in terms of Lie algebras) and those which provide solutions of Einstein-Maxwell equations are known from the literature. It is natural to ask which spaces from Komrakov's list provide solutions of the Einstein-Yang-Mills equations.
In the present paper, this question is studied in the special case of Lorentzian symmetric spaces. In particular, solutions with respect to an invariant connection in the bundle of orthonormal frames and a diagonal metric on the holonomy algebra are investigated. The ten cases with the nontrivial isotropy group are found.
Reviewer: Zdeněk Dušek (České Budějovice)Two-dimensional twistor manifolds and Teukolsky operators.https://zbmath.org/1459.830272021-05-28T16:06:00+00:00"Araneda, Bernardo"https://zbmath.org/authors/?q=ai:araneda.bernardoSummary: The Teukolsky equations are currently the leading approach for analysing stability of linear massless fields propagating in rotating black holes. It has recently been shown that the geometry of these equations can be understood in terms of a connection constructed from the conformal and complex structure of Petrov type D spaces. Since the study of linear massless fields by a combination of conformal, complex and spinor methods is a distinctive feature of twistor theory, and since versions of the twistor equation have recently been shown to appear in the Teukolsky equations, this raises the question of whether there are deeper twistor structures underlying this geometry. In this work we show that all these geometric structures can be understood naturally by considering a \textit{2-dimensional} twistor manifold, whereas in twistor theory the standard (projective) twistor space is 3-dimensional.\(g\)-Steiner, co-Steiner and normal points of bounded Euclidean submanifolds.https://zbmath.org/1459.520032021-05-28T16:06:00+00:00"Chen, Bang-Yen"https://zbmath.org/authors/?q=ai:chen.bang-yenThe Steiner point, aka the Steiner curvature centroid, was originally defined to be the geometric centroid of the system obtained by placing a mass equal to the magnitude of the exterior angle at each vertex of a triangle, being introduced by Jakob Steiner (1976--1863), a Swiss mathematician [ERAM 021.0662cj]. \textit{H. Flanders} [Mathematika 13, 181--188 (1966; Zbl 0146.17502)]
established that, for an even-dimensional convex closed hypersurface \(M^{n}\) in a Euclidean \(\left( n+1\right) \)-space \(\mathbb{E}^{n+1}\), the Steiner point \(s\left( M^{n}\right) \) is to be defined as
\[
s\left( M^{n}\right) =\frac{1}{c_{n}}\int\nolimits_{p\in M^{n}}\boldsymbol{x}K\left( p\right) dv
\]
where \(\boldsymbol{x}\) denotes the position vector field of \(M^{n}\) in \(\mathbb{E}^{n+1}\), \(dv\) is the volume element of \(M^{n}\), and \(K\left( p\right) \) denotes the Gauss-Kronecker curvature of \(M^{n}\) at a point \(p\in M^{n}\).
It is well known that the Steiner point by the hand of Flanders abides by the following properties [\textit{G. C. Shephard}, Can. J. Math. 18, 1294--1300 (1966; Zbl 0145.42801); J. Lond. Math. Soc. 43, 439--444 (1968; Zbl 0162.25801)].
\begin{itemize}
\item For any similar transformation \(a\), we have
\[
s\left( aM^{n}\right) =as\left( M^{n}\right).
\]
\item For any constant vector \(c\in\mathbb{E}^{n+1}\), we have
\[
s\left( M^{n}+c\right) =s\left( M^{n}\right) +c.
\]
\item \(s\left( M^{n}\right) \) is a continuous function of \(M^{n}\).
\item If \(\mathrm{\dim\,}M^{n}\) is positive, then \(s\left( M^{n}\right) \) is a relative interior point of \(M^{n}\).
\end{itemize}
This paper aims
\begin{itemize}
\item to extend the notion of Steiner points to the notion of \(g\)-Steiner points for bounded Euclidean submanifolds with arbitrary codimension,
\item to introduce the notions of co-Steiner and normal points for bounded Euclidean submanifolds via the notion of \(G\)-total curvature [\textit{B.-Y. Chen}, J. Differ. Geom. 7, 371--391 (1972; Zbl 0274.53060)],
\item to establish several fundamental properties for such points, and
\item to establish some links between \(g\)-Steiner, co-Steiner and normal points.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Polynomial symmetries of flat and homogeneous connections.https://zbmath.org/1459.530382021-05-28T16:06:00+00:00"Kal'nitskiĭ, V. S."https://zbmath.org/authors/?q=ai:kalnitskij.v-sSummary: The classification theorem is formulated for symmetries of homogeneous connections on smooth manifold. The cases admitting realization on manifolds are described.The tunneling effect for Schrödinger operators on a vector bundle.https://zbmath.org/1459.580122021-05-28T16:06:00+00:00"Klein, Markus"https://zbmath.org/authors/?q=ai:klein.markus"Rosenberger, Elke"https://zbmath.org/authors/?q=ai:rosenberger.elkeSummary: In the semiclassical limit \(\hbar \rightarrow 0\), we analyze a class of self-adjoint Schrödinger operators \(H_\hbar = \hbar^2 L + \hbar W + V\cdot\text{id}_{\mathcal{E}}\) acting on sections of a vector bundle \(\mathcal{E}\) over an oriented Riemannian manifold \(M\) where \(L\) is a Laplace type operator, \(W\) is an endomorphism field and the potential energy \(V\) has non-degenerate minima at a finite number of points \(m^1,\dots m^r \in M\), called potential wells. Using quasimodes of WKB-type near \(m^j\) for eigenfunctions associated with the low lying eigenvalues of \(H_\hbar\), we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension \(\ell + 1\). This dimension \(\ell\) determines the polynomial prefactor for exponentially small eigenvalue splitting.Collapsing of the line bundle mean curvature flow on Kähler surfaces.https://zbmath.org/1459.530692021-05-28T16:06:00+00:00"Takahashi, Ryosuke"https://zbmath.org/authors/?q=ai:takahashi.ryosukeLet \(X\) be a compact complex surface, \(\alpha\) a Kähler form (a closed Hermitian form), \(\hat F\) a closed real \((1,1)\)-form, and \(\psi_t\) a real-valued smooth function on \(X\) and solution of the line bundle mean curvature flow \(\displaystyle\frac{d\psi_t}{dt}=\Theta_\alpha-\hat\Theta\) such that \(\Theta_\alpha\) stands for the Lagrange phase and \(\Theta_\alpha\equiv\hat\Theta\ (2\pi)\). Then, by assuming that \(\psi_0\) is hypercritical, \(\cot\hat\Theta_\alpha+\hat F\ge 0\), and \(\displaystyle\hat\Theta\ge \frac{\pi}{2}\), the author deduces that there is a family of curves \((C_j)_{j\in\mathcal{J}}\) (\(\mathcal{J}\) is a finite subset of \(\mathbb N\)) with negative self-intersection such that \(\displaystyle\lim_{t\to\infty}\psi_t=\psi_\infty\in C_{\mathrm{loc}}^\infty(X\setminus\cup_{j\in\mathcal{J}}C_j)\) and \(F_\infty=\hat F+\sqrt{-1}\partial\overline\partial\psi_\infty\) is a smooth Kähler current on \(X\setminus\cup_{j\in\mathcal J} C_j\) and satisfying \(\Im\left(\exp(-\sqrt{-1}\hat\Theta)(\alpha+\sqrt{-1}F_{\psi_\infty})^2\right)=0\) (Theorem 1).
Reviewer: Mohammed El Aïdi (Bogotá)Lagrangian submanifolds of the nearly Kähler 6-sphere and Chen's equality.https://zbmath.org/1459.530312021-05-28T16:06:00+00:00"Sharma, Ramesh"https://zbmath.org/authors/?q=ai:sharma.rameshSummary: We obtain the following results: (i) If the normal curvature operator of a Lagrangian submanifold \(M\) of the nearly Kähler \(S^6\) (with almost complex structure \(J)\) annihilates a vector field \(V\) normal to \(M\), then the shape operator with respect to \(V\) vanishes, and \(M\) is quasi-Einstein and is Sasakian if \(JV\) is Killing on \(M\). (ii) If a Lagrangian submanifold \(M\) of the nearly Kähler \(S^6\) satisfies the Chen's equality and is conformally flat, then it is the totally geodesic \(S^3\). Finally we prove the result: (iii) If the almost contact metric structure induced by a unit tangent vector field on a Lagrangian submanifold of the nearly Kähler \(S^6\) is normal, then it is Sasakian.
For the entire collection see [Zbl 1454.53006].Spaces of harmonic maps of the projective plane to the four-dimensional sphere.https://zbmath.org/1459.530652021-05-28T16:06:00+00:00"Gabdurakhmanov, Ravil"https://zbmath.org/authors/?q=ai:gabdurakhmanov.ravilThe paper investigates the spaces of harmonic maps of the projective plane to the four-dimensional sphere, by means of twistor lifts. It is shown that such spaces are empty in the case of even harmonic degree. In the case of harmonic degree less than 6 it is proved that such spaces are path-connected and an explicit parameterization of the canonical representatives is found. Moreover, comparisons with the known results for harmonic maps of the two-dimensional sphere to the four-dimensional sphere of harmonic degree less than 6 are provided.
Reviewer: Vladimir Balan (Bucureşti)Notes about a new metric on the cotangent bundle.https://zbmath.org/1459.530262021-05-28T16:06:00+00:00"Ocak, Filiz"https://zbmath.org/authors/?q=ai:ocak.filizSummary: In this article, we construct a new metric \(\breve G=^R\nabla+\sum^m _{i,j=1}a^{ji}\delta p_j\delta p_i\) in the cotangent bundle, where \(^R\nabla\) is the Riemannian extension and \(a^{ji}\) is a symmetric \((2,0)\)-tensor field on a differentiable manifold.Slant helices that constructed from hyperspherical curves in the \(n\)-dimensional Euclidean space.https://zbmath.org/1459.530092021-05-28T16:06:00+00:00"Altunkaya, Bülent"https://zbmath.org/authors/?q=ai:altunkaya.bulentSummary: In this work, we study slant helices in the \(n\)-dimensional Euclidean space. We give methods to determine the position vectors of slant helices from arclength parameterized curves that lie on the unit hypersphere. By means of these methods, first we characterize slant helices and Salkowski curves which lie on \(2n\)-dimensional hyperboloid. After that, we characterize rectifying slant helices which are geodesics of \(2n\)-dimensional cone.On the geometry of some \(( \alpha, \beta )\)-metrics on the nilpotent groups \(H(p, r)\).https://zbmath.org/1459.530712021-05-28T16:06:00+00:00"Nejadahmad, Masumeh"https://zbmath.org/authors/?q=ai:nejadahmad.masumeh"Moghaddam, Hamid Reza Salimi"https://zbmath.org/authors/?q=ai:salimi-moghaddam.hamid-rezaSummary: In this paper we study the Riemann-Finsler geometry of the Lie groups \(H(p, r)\) which are a generalization of the Heisenberg Lie groups. For a certain Riemannian metric \(\langle\cdot,\cdot\rangle\), the Levi-Civita connection and the sectional curvature are given. We classify all left invariant Randers metrics of Douglas type induced by \(\langle\cdot,\cdot\rangle\), compute their flag curvatures and show that all of them are non-Berwaldian.Curvature properties of quasi-para-Sasakian manifolds.https://zbmath.org/1459.530422021-05-28T16:06:00+00:00"Erken, İ. Küpeli"https://zbmath.org/authors/?q=ai:kupeli-erken.i|erken.irem-kupeliSummary: The paper is devoted to study quasi-para-Sasakian manifolds. Basic properties of such manifolds are obtained and general curvature identities are investigated. Next it is proved that if \(M\) is a quasi-para-Sasakian manifold of constant curvature \(K\). Then \(K\leqslant 0\) and \((i)\) if \(K= 0\), the manifold is paracosymplectic, \((ii)\) if \(K <0\), the quasi-para-Sasakian structure of \(M\) is obtained by a homothetic deformation of a para-Sasakian structure. Finally, an example of a 3-dimensional proper quasi-para-Sasakian manifold is constructed.On Hopf hypersurfaces of the homogeneous nearly Kähler \(S^3 \times S^3\).https://zbmath.org/1459.530292021-05-28T16:06:00+00:00"Hu, Zejun"https://zbmath.org/authors/?q=ai:hu.zejun"Yao, Zeke"https://zbmath.org/authors/?q=ai:yao.zekeThe authors study Hopf hypersurfaces \(M\) in the homogeneous nearly Kähler manifold \(S^3 \times S^3\). They prove that there does not exist such an \(M\) admitting two distinct principal curvatures. The paper contains an important classification result: If \(M\) has three distinct principal curvatures and if the holomorphic distribution \(\{ U \}^{\perp}\) associated to the structure vector field is preserved by the (canonical) almost product structure \(P\), then \(M\) is given, locally, by one of three explicit embeddings of \(S^3 \times S^2\) into \(S^3 \times S^3\).
Reviewer: Gabriel Teodor Pripoae (Bucureşti)Douglas-square metrics with vanishing mean stretch curvature.https://zbmath.org/1459.530362021-05-28T16:06:00+00:00"Tayebi, Akbar"https://zbmath.org/authors/?q=ai:tayebi.akbar"Izadian, Neda"https://zbmath.org/authors/?q=ai:izadian.nedaSummary: In this paper, we consider the class of square metrics \(F= \alpha+2\beta+\beta^2/\alpha\) where \(\alpha=\sqrt{a_{ij}y^iy^j}\) is a Riemannian metric and \(\beta=b_i(x)y^i\) is a one-form on a manifold \(M\). Let \((M, F)\) be a Douglas-square manifold. We show that \(F\) is a Berwald metric if and only if it a weakly stretch metric. It results that, a Douglas-square metric is \(R\)-quadratic if and only if it is a Berwald metric.Eigenvalue estimates for multi-form modified Dirac operators.https://zbmath.org/1459.353222021-05-28T16:06:00+00:00"Gutowski, Jan"https://zbmath.org/authors/?q=ai:gutowski.jan-b"Papadopoulos, George"https://zbmath.org/authors/?q=ai:papadopoulos.georgeSummary: We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a \(k\)-degree form \(0\leq k\leq 4\), those modified with multi-degree \((0,k)\)-form \(0\leq k\leq 3\) and the horizon Dirac operators which are modified with a multi-degree \((1,2,4)\)-form. In particular, we give the necessary geometric conditions for such operators to admit zero modes as well as those for the zero modes to be parallel with a respect to a suitable connection. We also demonstrate that manifolds which admit such parallel spinors are associated with twisted covariant form hierarchies which generalize the conformal Killing-Yano forms.Optimal bang-bang trajectories in sub-Finsler problems on the Engel group.https://zbmath.org/1459.530352021-05-28T16:06:00+00:00"Sachkov, Yuriĭ"https://zbmath.org/authors/?q=ai:sachkov.yuri-lSummary: The Engel group is the four-dimensional nilpotent Lie group of step 3, with 2 generators. We consider a one-parameter family of left-invariant rank 2 sub-Finsler problems on the Engel group with the set of control parameters given by a square centered at the origin and rotated by an arbitrary angle. We adopt the viewpoint of time-optimal control theory. By Pontryagin's maximum principle, all sub-Finsler length minimizers belong to one of the following types: abnormal, bang-bang, singular, and mixed. Bang-bang controls are piecewise controls with values in the vertices of the set of control parameters.
We describe the phase portrait for bang-bang extremals.
In previous work, it was shown that bang-bang trajectories with low values of the energy integral are optimal for arbitrarily large times. For optimal bang-bang trajectories with high values of the energy integral, a general upper bound on the number of switchings was obtained.
In this paper we improve the bounds on the number of switchings on optimal bang-bang trajectories via a second-order necessary optimality condition due to A. Agrachev and R. Gamkrelidze. This optimality condition provides a quadratic form, whose sign-definiteness is related to optimality of bang-bang trajectories. For each pattern of these trajectories, we compute the maximum number of switchings of optimal control. We show that optimal bang-bang controls may have not more than 9 switchings. For particular patterns of bang-bang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous work.
On the basis of the results of this work we can start to study the cut time along bang-bang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent work.On certain classes of \(\mathbf{Sp}(4, \mathbb{R})\) symmetric \(G_2\) structures.https://zbmath.org/1459.530392021-05-28T16:06:00+00:00"Nurowski, Paweł"https://zbmath.org/authors/?q=ai:nurowski.pawelSummary: We find two different families of \(\mathbf{Sp}(4, \mathbb{R})\) symmetric \(G_2\) structures in seven dimensions. These are \(G_2\) structures with \(G_2\) being the split real form of the simple exceptional complex Lie group \(G_2\). The first family has \(\tau_2 \equiv 0\), while the second family has \(\tau_1 \equiv \tau_2 \equiv 0\), where \(\tau_1, \tau_2\) are the celebrated \(G_2\)-invariant parts of the intrinsic torsion of the \(G_2\) structure. The families are different in the sense that the first one lives on a homogeneous space \(\mathbf{Sp}(4, \mathbb{R})/\mathbf{SL}(2, \mathbb{R})_l\), and the second one lives on a homogeneous space \(\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s\). Here \(\mathbf{SL}(2,\mathbb{R})_l\) is an \(\mathbf{SL}(2,\mathbb{R})\) corresponding to the \(\mathfrak{sl}(2,\mathbb{R})\) related to the long roots in the root diagram of \(\mathfrak{sp}(4,\mathbb{R})\), and \(\mathbf{SL}(2,\mathbb{R})_s\) is an \(\mathbf{SL}(2,\mathbb{R})\) corresponding to the \(\mathfrak{sl}(2,\mathbb{R})\) related to the short roots in the root diagram of \(\mathfrak{sp}(4,\mathbb{R})\).The Yamabe invariants of Inoue surfaces, Kodaira surfaces, and their blowups.https://zbmath.org/1459.530672021-05-28T16:06:00+00:00"Albanese, Michael"https://zbmath.org/authors/?q=ai:albanese.michaelSummary: Shortly after the introduction of Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a compact Kähler surface is determined by its Kodaira dimension. In this paper, we show that LeBrun's Theorem is no longer true for non-Kähler surfaces. In particular, we show that the Yamabe invariants of Inoue surfaces and their blowups are all zero. We also take this opportunity to record a proof that the Yamabe invariants of Kodaira surfaces and their blowups are all zero, as previously indicated by LeBrun.Bounding the invariant spectrum when the scalar curvature is non-negative.https://zbmath.org/1459.530472021-05-28T16:06:00+00:00"Hall, Stuart J."https://zbmath.org/authors/?q=ai:hall.stuart-james"Murphy, Thomas"https://zbmath.org/authors/?q=ai:murphy.thomas-e|murphy.thomas-j|murphy.thomas-brendan|murphy.thomas-w-jun|murphy.thomas-n|murphy.thomas-mSummary: On compact Riemannian manifolds with a large isometry group we investigate the invariant spectrum of the ordinary Laplacian. For either a toric Kähler metric, or a rotationally-symmetric metric on the sphere, we produce upper bounds for all eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
For the entire collection see [Zbl 1454.53006].On six-dimensional AH-submanifolds of class \(W_1 \oplus W_2 \oplus W_4\) in Cayley algebra.https://zbmath.org/1459.530412021-05-28T16:06:00+00:00"Banaru, G. A."https://zbmath.org/authors/?q=ai:banaru.galina-anatolevnaIn the well-known paper [\textit{A. Gray} and \textit{L. M. Hervella}, Ann. Mat. Pura Appl. (4) 123, 35--58 (1980; Zbl 0444.53032)], sixteen classes of almost Hermitian structures are defined. In this note, the author studies the class \(W_1\oplus W_2\oplus W_4\) containing all Kählerian, nearly Kählerian, almost Kählerian, locally conformal Kählerian, quasi-Kählerian and Vaisman-Gray manifolds.
The author considers an almost Hermitian structure of the class \(W_1\oplus W_2\oplus W_4\), induced on six-dimensional submanifolds of the Cayley algebra. These six-dimensional submanifolds are equipped with an almost Hermitian structure of the class \(W_1\oplus W_2\oplus W_4\) defined by means of three-fold vector cross products. The Cartan structural equations of the \(W_1\oplus W_2\oplus W_4\)-structure on such six-dimensional submanifolds of the octave algebra are obtained. It is proved that, if a six-dimensional \(W_1\oplus W_2\oplus W_4\)-submanifold of the Cayley algebra satisfies the quasi-Sasakian hypersurfaces axiom (i.e., a hypersurface with a quasi-Sasakian structure passes through every point of such submanifold), then it is an almost Kählerian manifold. It is also proved that, if a six-dimensional \(W_1\oplus W_2\oplus W_4\)-submanifold of the Cayley algebra satisfies the eta-quasi-umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kählerian manifold.
Reviewer: Nicolai K. Smolentsev (Kemerovo)Properly embedded surfaces with prescribed mean curvature in \(\mathbb{H}^2 \times \mathbb{R}\).https://zbmath.org/1459.530602021-05-28T16:06:00+00:00"Bueno, Antonio"https://zbmath.org/authors/?q=ai:bueno.antonioSummary: The aim of this paper is to extend classic results of the theory of constant mean curvature surfaces in the product space \(\mathbb{H}^2 \times \mathbb{R}\) to the class of immersed surfaces whose mean curvature is given as a \(C^1\) function depending on their angle function. We cover topics such as the existence of a priori curvature and height estimates for graphs and a structure-type result, which classifies properly embedded surfaces with finite topology and at most one end.Generalized quivers, orthogonal and symplectic representations, and Hitchin-Kobayashi correspondences.https://zbmath.org/1459.530682021-05-28T16:06:00+00:00"de Araujo, Artur"https://zbmath.org/authors/?q=ai:de-araujo.arturA quiver \(Q\) is a finite directed graph with a set of arrows and two maps, which assign to each arrow its head and tail, respectively; a representation of \(Q\) is an assignment of an object for each vertex and of a morphism for each arrow. A twisted quiver associated to a
Kähler manifold \(X\) allows the choice of a vector bundle over \(X\) for each arrow in \(Q\); a twisted quiver bundle representation of \(Q\) is a choice of a vector bundle for each vertex and of a morphism for each arrow.
The author studies quiver bundles with additional symmetries: he generalizes the approach by \textit{H. Derksen} and \textit{J. Weyman} [Colloq. Math. 94, No. 2, 151--173 (2002; Zbl 1025.16010)] to arbitrary reductive groups of symmetries \(G\); when \(G\) is the orthogonal or the symplectic group, he proves that representations for such generalized quivers correspond to representations of symmetric quivers; he studies supermixed quivers and characterizes the polystable forms of such representations; he discusses the Hitchin-Kobayashi correspondences for these objects. The paper ends with a section of examples providing geometrical interpretations of generalized quivers.
Reviewer: Gabriel Teodor Pripoae (Bucureşti)Branching geodesics in sub-Riemannian geometry.https://zbmath.org/1459.530442021-05-28T16:06:00+00:00"Mietton, Thomas"https://zbmath.org/authors/?q=ai:mietton.thomas"Rizzi, Luca"https://zbmath.org/authors/?q=ai:rizzi.lucaGeodesics in sub-Riemannian manifolds are not yet well understood. For example, it is not known whether or not sub-Riemannian geodesics must even be \(C^1\).
A geodesic \(\gamma:[0,1] \to M\) in a manifold \(M\) is said to branch at time \(t \in (0,1)\) if there is another geodesic \(\gamma'\) with \(\gamma|_{[0,t]} = \gamma'|_{[0,t]}\) but \(\gamma|_{[0,s]} \neq \gamma'|_{[0,s]}\) for any \(s>t\). In a Riemannian manifold with a lower curvature bound, it is known that all geodesics are non-branching. In contrast, it is shown in the paper under review that normal geodesics in sub-Riemannian manifolds may indeed branch. Recall that a sub-Riemannian geodesic is normal if its associated control is not a critical point of the end-point map.
The authors observe in Theorem 5 that a normal geodesic \(\gamma\) branches at time \(t\) if and only if the corank function is discontinuous at \(t\). The corank function assigns to any time \(t\) the corank of \(D_{u_t}E_x\) where \(u_t\) is the minimal control of \(\gamma|_{[0,t]}\) and \(E_x\) is the endpoint map. Further, the authors prove in Theorem 13 that any non-increasing, left continuous function \(f:[0,1] \to \mathbb{N}\) is the corank function of a normal sub-Riemannian geodesic, and thus this geodesic must branch at the discontinuities of \(f\).
They also show in Theorem~7 that the size of the time interval around \(t\) over which some compact family of branching paths is length minimizing is uniformly controlled.
Finally, in Section 4, the authors provide an example of a sub-Riemannian structure on \(\mathbb{R}^3\) for which the path \(t \mapsto (0,t,0)\) branches at \(t=0\).
Reviewer: Scott Zimmerman (Marion)Heat content asymptotics for sub-Riemannian manifolds.https://zbmath.org/1459.353642021-05-28T16:06:00+00:00"Rizzi, Luca"https://zbmath.org/authors/?q=ai:rizzi.luca"Rossi, Tommaso"https://zbmath.org/authors/?q=ai:rossi.tommasoSummary: We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the full asymptotic series:
\[
Q_{\Omega}(t)=\sum_{k=0}^\infty a_k t^{k/2},\text{ as }t\to 0.
\]
We compute explicitly the coefficients up to order \(k=5\), in terms of sub-Riemannian invariants of the domain. Furthermore, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension. As a particular case we recover, using non-probabilistic techniques, the order 2 formula recently obtained by \textit{J. Tyson} and \textit{J. Wang} [Commun. Partial Differ. Equations 43, No. 3, 467--505 (2018; Zbl 1391.53043)] in the Heisenberg group. A consequence of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points. In particular, we prove that the higher order coefficients in the asymptotics can blow-up in their presence. A key tool for this last result is an exact formula for the distance from a specific surface with an isolated characteristic point in the Heisenberg group, which is of independent interest.On contact pseudo-metric manifolds satisfying a nullity condition.https://zbmath.org/1459.530752021-05-28T16:06:00+00:00"Ghaffarzadeh, Narges"https://zbmath.org/authors/?q=ai:ghaffarzadeh.narges"Faghfouri, Morteza"https://zbmath.org/authors/?q=ai:faghfouri.mSummary: In this paper, we aim to introduce and study \((\kappa, \mu)\)-contact pseudo-metric manifold and prove that if the \(\varphi \)-sectional curvature of any point of \(M\) is independent of the choice of \(\phi \)-section at the point, then it is constant on \(M\) and accordingly the curvature tensor. Also, we introduce generalized \((\kappa, \mu)\)-contact pseudo-metric manifold and prove for \(n > 1\), that a non-Sasakian generalized \((\kappa, \mu)\)-contact pseudo-metric manifold is a \((\kappa, \mu)\)-contact pseudo-metric manifold.The measure preserving isometry groups of metric measure spaces.https://zbmath.org/1459.530462021-05-28T16:06:00+00:00"Guo, Yifan"https://zbmath.org/authors/?q=ai:guo.yifanSummary: Bochner's theorem says that if \(M\) is a compact Riemannian manifold with negative Ricci curvature, then the isometry group \(\text{Iso}(M)\) is finite. In this article, we show that if \((X, d, m)\) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure preserving isometry group \(\text{Iso} (X, d, m)\) is finite. We also give an effective estimate on the order of the measure preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Émery Ricci curvature except for small portions.Finite-time degeneration for variants of Teichmüller harmonic map flow.https://zbmath.org/1459.530152021-05-28T16:06:00+00:00"Robertson, Craig"https://zbmath.org/authors/?q=ai:robertson.craig"Rupflin, Melanie"https://zbmath.org/authors/?q=ai:rupflin.melanieTeichmüller harmonic map flow is a geometric flow that is designed to flow surfaces to minimal surfaces. In this paper, the authors consider the question of whether solutions of variants of the Teichmüller harmonic map flow from surfaces \(M\) to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least \(2\), as well as the flow from cylinders, they prove that such a finite-time degeneration must occur in situations where the image of thin collars is `stretching out' at a rate of at least \(\mbox{inj}(M,g)^{-(\frac{1}{4}+\delta)}\), and they construct targets in which the flow from cylinders must indeed degenerate in finite time. For the rescaled Teichmüller harmonic map flow, the condition that the image stretches out is not only sufficient but also necessary and they prove the following sharp result: Solutions of the rescaled flow cannot degenerate in finite time if the image stretches out at a rate of no more than \(\vert\log(\mbox{inj}(M,g))\vert^{\frac{1}{2}}\), but must degenerate in finite time if it stretches out at a rate of at least \(\vert\log(\mbox{inj}(M,g))\vert^{\frac{1}{2}+\delta}\) for some \(\delta>0\).
Reviewer: Atsushi Fujioka (Osaka)Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds.https://zbmath.org/1459.580132021-05-28T16:06:00+00:00"Braun, Mathias"https://zbmath.org/authors/?q=ai:braun.mathias"Habermann, Karen"https://zbmath.org/authors/?q=ai:habermann.karen"Sturm, Karl-Theodor"https://zbmath.org/authors/?q=ai:sturm.karl-theodorSummary: Given a metric measure space \((X,\mathsf{d},\mathfrak{m})\) and a lower semicontinuous, lower bounded function \(k:X\to\mathbb{R}\), we prove the equivalence of the synthetic approaches to Ricci curvature a \(x\in X\) being bounded from below by \(k(x)\) in terms of
\begin{itemize}
\item the Bakry-Émery estimate \(\Delta\Gamma(f)/2-\Gamma(f,\Delta f) \geq k\Gamma(f)\) in an appropriate weak formulation, and
\item the curvature-dimension condition \(\operatorname{CD}(k,\infty)\) in the sense of Lott-Sturm-Villani with variable \(k\).
\end{itemize}
Moreover, for all \(p\in(1,\infty)\), these properties hold if and only if the perturbed \(p\)-transport cost
\[
W^{\underline{k}}_p(\mu_1,\mu_2,t):=\inf\limits_{(\mathsf{b}^1,\mathsf{b}^2)} \mathbb{E}[\operatorname{e}^{\int_0^{2t}p\underline{k}(\mathsf{b}_r^1,\mathsf{b}_r^2)/2 \operatorname{d}r}\mathsf{d}^p(\mathsf{b}_{2t}^1,\mathsf{b}_{2t}^2)]^{1/p}
\]
is nonincreasing in \(t\). The infimum here is taken over pairs of coupled Brownian motions \(\mathsf{b}^1\) and \(\mathsf{b}^2\) on \(X\) with given initial distributions \(\mu_1\) and \(\mu_2\), respectively, and \(\underline{k}(x,y):=\inf_\gamma\int_0^1k(\gamma_s)\operatorname{d}s\) denotes the ``average'' of \(k\) along geodesics \(\gamma\) connecting \(x\) and \(y\).
Furthermore, for any pair of initial distributions \(\mu_1\) and \(\mu_2\) on \(X\), we prove the existence of a pair of coupled Brownian motions \(\mathsf{b}^1\) and \(\mathsf{b}^2\) such that a.s. for every \(s,t\in[0,\infty)\) with \(s\leq t\), we have
\[
\mathsf{d}(\mathsf{b}_t^1,\mathsf{b}_t^2)\leq\operatorname{e}^{-\int_s^t\underline{k}(\mathsf{b}_r^1,\mathsf{b}_r^2)/2\operatorname{d}r}\mathsf{d}(\mathsf{b}_s^1,\mathsf{b}_s^2).
\]Small gauge transformations and universal geometry in heterotic theories.https://zbmath.org/1459.530372021-05-28T16:06:00+00:00"McOrist, Jock"https://zbmath.org/authors/?q=ai:mcorist.jock"Sisca, Roberto"https://zbmath.org/authors/?q=ai:sisca.robertoSummary: The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is 'holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in \(\alpha^{\backprime}\) and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a choice of holomorphic structure and Lee form on a 'universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.Concircular curvature on warped product manifolds and applications.https://zbmath.org/1459.530662021-05-28T16:06:00+00:00"De, Uday Chand"https://zbmath.org/authors/?q=ai:de.uday-chand"Shenawy, Sameh"https://zbmath.org/authors/?q=ai:shenawy.sameh"Ünal, Bülent"https://zbmath.org/authors/?q=ai:unal.bulentOn a (pseudo-)Riemannian manifold \(M\), there is a concircular curvature tensor which is invariant under concircular transformations. The manifold is called concircularly flat if its concircular curvature tensor vanishes at every point. A concircularly flat manifold \(M\) is of constant curvature. In this sense, the concircular curvature measures the deviation of \(M\) from constant curvature.
In the paper under review, the authors study the concircular curvature tensor on warped product manifolds. In particular, they study
divergence-free concircular curvature tensor on these manifolds. They obtain applications to two different \(n\)-dimensional spacetimes, namely, generalized Robertson-Walker spacetimes and standard static spacetimes.
Reviewer: Athanase Papadopoulos (Strasbourg)Rank of ordinary webs in codimension one an effective method.https://zbmath.org/1459.530222021-05-28T16:06:00+00:00"Dufour, Jean-Paul"https://zbmath.org/authors/?q=ai:dufour.jean-paul"Lehmann, Daniel"https://zbmath.org/authors/?q=ai:lehmann.daniel-j|lehmann.danielSummary: We are interested by holomorphic \(d\)-webs \(W\) of codimension one in a complex \(n\)-dimensional manifold \(M\). If they are \textit{ordinary}, i.e. if they satisfy to some condition of genericity (whose precise definition is recalled below), we proved in [CL] that their rank \(\rho (W)\) is upper-bounded by a certain number \(\pi^\prime (n, d)\) (which, for \(n\geq 3\), is strictly smaller than the Castelnuovo-Chern's bound \(\pi(n,d))\).
In fact, denoting by \(c(n,h)\) the dimension of the space of homogeneous polynomials of degree \(h\) with \(n\) unknowns, and by \(h_0\) the integer such that
\[
c(n, h_0-1)<d\leq c(n, h_0),
\]
\(\pi'(n,d)\) is just the first number of a decreasing sequence of positive integers
\[
\pi'(n,d)=\rho_{h_0-2}\geq\rho_{h_0-1}\geq\cdots\geq\rho_h\geq\rho_{h+1} \geq\cdots\geq\rho_\infty=\rho(W)\geq 0
\]
becoming stationary equal to \(\rho (W)\) after a finite number of steps. This sequence is an interesting invariant of the web, refining the data of the only rank.
The method is effective: theoretically, we can compute \(\rho_h\) for any given \(h\); and, as soon as two consecutive such numbers are equal \((\rho_h=\rho_{h+1} ,h\geq h_0-2)\), we can construct a holomorphic vector bundle \(R_h \to M\) of rank \(\rho_h\), equipped with a tautological holomorphic connection \(\nabla^h\) whose curvature \(K^h\) vanishes iff the above sequence is stationary from there. Thus, we may stop the process at the first step where the curvature vanishes, and compute the rank without to have to exhibit explicitly independant abelian relations.
Examples will be given.Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres.https://zbmath.org/1459.530642021-05-28T16:06:00+00:00"Van der Veken, Joeri"https://zbmath.org/authors/?q=ai:van-der-veken.joeri"Wijffels, Anne"https://zbmath.org/authors/?q=ai:wijffels.anneSummary: The Gauss map of a hypersurface of a unit sphere \(S^{n+1}(1)\) is a Lagrangian immersion into the complex quadric \(Q^n\) and, conversely, every Lagrangian submanifold of \(Q^n\) is locally the image under the Gauss map of several hypersurfaces of \(S^{n+1}(1)\). In this paper, we give explicit constructions for these correspondences and we prove a relation between the principal curvatures of a hypersurface of \(S^{n+1}(1)\) and the local angle functions of the corresponding Lagrangian submanifold of \(Q^n\). The existence of such a relation is remarkable since the definition of the angle functions depends on the choice of an almost product structure on \(Q^n\) and since several hypersurfaces of \(S^{n+1}(1)\), with different principal curvatures, correspond to the same Lagrangian submanifold of \(Q^n\).
For the entire collection see [Zbl 1454.53006].Sesquilinear forms and symmetric spaces.https://zbmath.org/1459.530562021-05-28T16:06:00+00:00"Thorbergsson, Gudlaugur"https://zbmath.org/authors/?q=ai:thorbergsson.gudlaugurSummary: Let \(f\) be a sesquilinear form on \(\mathbb{F}^n\) with positive Witt index \(r\) where \(\mathbb{F}\) is \(\mathbb{R}\), \(\mathbb{C}\), or \(\mathbb{H}\). Let \(N_i(\mathbb{F}^n,f)\) denote the space of \(i\)-dimensional totally isotropic subspaces of \(\mathbb{F}^n\) with respect to \(f\) where \(i\leq r\). Then our main result will be the observation that \(N_i(\mathbb{F}^n,f)\) is a symmetric space if and only if \(n=2i\). This gives us seven series of compact symmetric spaces. If we add the three series of Grassmannians \(G_i(\mathbb{F}^n)\) over \(\mathbb{F}\), we get all ten series of classical compact symmetric spaces.
For the entire collection see [Zbl 1454.53006].Warped product hypersurfaces in pseudo-Riemannian real space forms.https://zbmath.org/1459.530302021-05-28T16:06:00+00:00"Moruz, Marilena"https://zbmath.org/authors/?q=ai:moruz.marilena"Vrancken, Luc"https://zbmath.org/authors/?q=ai:vrancken.lucSummary: We study hypersurfaces in pseudo-Riemannian real space forms of non-zero sectional curvature, which write as a warped product of a \(1\)-dimensional base with an \((n-1)\)-manifold of constant sectional curvature. We show that either they have constant sectional curvature or they are contained in a rotational hypersurface. Therefore, we first define rotational hypersurfaces in indefinite real space forms of non-zero sectional curvature.
For the entire collection see [Zbl 1454.53006].On isoparametric linear Weingarten hypersurfaces in Riemannian and Lorentzian space forms.https://zbmath.org/1459.530632021-05-28T16:06:00+00:00"Özgür, Cihan"https://zbmath.org/authors/?q=ai:ozgur.cihanSummary: Linear Weingarten hypersurfaces have been extensively studied by various aspects in the literature. In the present paper, we deal with the classifications of isoparametric linear Weingarten hypersurfaces in Riemannian and Lorentzian space forms.
For the entire collection see [Zbl 1454.53006].Rigidity at infinity for lattices in rank-one Lie groups.https://zbmath.org/1459.530502021-05-28T16:06:00+00:00"Savini, Alessio"https://zbmath.org/authors/?q=ai:savini.alessioThe author studies rigidity properties for lattices in the group \(G_p=\mathrm{PU}(p,1)\) or \(\mathrm{PS}p(p,1)\). The case of \(\mathrm{PO}(p,1)\) has been considered by \textit{M. Bucher} et al. [Springer INdAM Ser. 3, 47--76 (2013; Zbl 1268.53056)] and also by \textit{S. Francaviglia} and \textit{B. Klaff} [Geom. Dedicata 117, 111--124 (2006; Zbl 1096.51004)]. This study originates in the results of Mostow about rigidity properties of lattices in locally symmetric spaces. Let \(X^p\) be the hyperbolic space associated to \(G_p\) and \(\Gamma \subset G_p\) a lattice such that the manifold \(M^p=\Gamma \backslash X^p\) is non-compact and has finite volume. For a representation \(\rho \) of \(\Gamma \) in \(G_m\), \(m\geq p\), the volume \(\mathrm{Vol}(\rho )\) is defined as the infimum of volumes of submanifolds in \(X^m\) associated to certain \(\rho \)-equivariant maps. Assume that \(\Gamma \) is without torsion. It is proven that \(\mathrm{Vol}(\rho )\leq\mathrm{Vol}(M^p)\) and equality holds if the representation \(\rho \) is a faithful representation of \(\Gamma \) into the isometry group of a totally geodesic copy of \(X^p\) contained in \(X^m\). Furthermore, the author considers a sequence \(\rho _n\) of representations of \(\Gamma \) in \(G_m\) such that
\[\lim _{n\to \infty }\mathrm{Vol}(\rho _n)=\mathrm{Vol}(M^p).\]
Then it is shown that there are \(g_n\in G_m\) such that \(g_n\circ \rho _n\circ g_n^{-1}\)
converges to a representation which preserves a totally geodesic copy of \(X^p\) and whose \(X^p\) component is conjugated to the standard lattice embedding of \(\Gamma \) into \(G_p\).
Reviewer: Jacques Faraut (Paris)Gravitational instantons of type \(D_k\) and a generalization of the Gibbons-Hawking ansatz.https://zbmath.org/1459.830122021-05-28T16:06:00+00:00"Ionaş, Radu A."https://zbmath.org/authors/?q=ai:ionas.radu-aSummary: We describe a quaternionic-based Ansatz generalizing the Gibbons-Hawking Ansatz to a class of hyperkähler metrics with hidden symmetries. We then apply it to obtain explicit expressions for gravitational instanton metrics of type \(D_k\).Rigidity and stability estimates for minimal submanifolds in the hyperbolic space.https://zbmath.org/1459.530592021-05-28T16:06:00+00:00"Bezerra, A. C."https://zbmath.org/authors/?q=ai:bezerra.adriano-c"Manfio, F."https://zbmath.org/authors/?q=ai:manfio.fernandoSummary: In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold \(M^n\) in the hyperbolic space \(\mathbb{H}^{n + m}\) in order to show that \(M^n\) is totally geodesic. We also obtain sharp upper bound estimates for the first eigenvalue of the super stability operator in the case of \(M\) is a surface in \(\mathbb{H}^4\).Statistical manifolds and their submanifolds. Results on Chen-like invariants.https://zbmath.org/1459.530582021-05-28T16:06:00+00:00"Mihai, Ion"https://zbmath.org/authors/?q=ai:mihai.ion-alexandru|mihai.ionSummary: Statistical manifolds were introduced by \textit{S.-i. Amari} [Differential-geometrical methods in statistics. Corr. 2nd. printing. Berlin etc.: Springer-Verlag (1990; Zbl 0701.62008)]. They generalize the Hessian manifolds. The geometry of statistical manifolds and their submanifolds is an actual topic of research in pure and applied mathematics. M. E. Aydin, A. Mihai and the present author [\textit{M. E. Aydin} et al., Filomat 29, No. 3, 465--477 (2015; Zbl 06749018)] obtained geometric inequalities for the scalar curvature and Ricci curvature associated to the dual connections for submanifolds in statistical manifolds of constant curvature. The same authors [\textit{M. E. Aydin} et al., Bull. Math. Sci. 7, No. 1, 155--166 (2017; Zbl 1371.53053)] proved a generalized Wintgen inequality for such submanifolds. Recently, in co-operation with \textit{A. Mihai} Mathematics 6, No. 3, Paper No. 44, 8 p. (2018; Zbl 1393.53046)], we initiated the study of the geometry of submanifolds in Hessian manifolds of constant Hessian curvature and established a Euler inequality and a Chen-Ricci inequality for such submanifolds. We continue the study of Chen-like invariants on submanifolds in Hessian manifolds of constant Hessian curvature.
For the entire collection see [Zbl 1454.53006].On the geometry of Einstein spaces: a note on their curvature symmetries.https://zbmath.org/1459.530522021-05-28T16:06:00+00:00"Mihai, Adela"https://zbmath.org/authors/?q=ai:mihai.adelaSummary: We recall few justifications for the study and therefore for the definition of an Einstein manifold. Recent results from the author and U. Simon [\textit{A. Mihai} and \textit{U. Simon}, Colloq. Math. 152, No. 1, 23--28 (2018; Zbl 1409.53043)]on curvature symmetries on Einstein spaces of arbitrary dimension \(n\geq 4\) are given.
For the entire collection see [Zbl 1454.53006].On deformation quantization using super twistorial double fibration.https://zbmath.org/1459.530792021-05-28T16:06:00+00:00"Hirota, Yuji"https://zbmath.org/authors/?q=ai:hirota.yuji"Miyazaki, Naoya"https://zbmath.org/authors/?q=ai:miyazaki.naoya"Taniguchi, Tadashi"https://zbmath.org/authors/?q=ai:taniguchi.tadashiSummary: We study deformation quantization for a complex supermanifold. Taking up a super twistor space whose body is a Calabi-Yau manifold concretely, we construct a double fibration and demonstrate that a certain super Calabi-Yau twistor space is deformation quantizable via the double fibration.
For the entire collection see [Zbl 1433.53003].Sasakian manifolds satisfying certain conditions on \(Q\) tensor.https://zbmath.org/1459.530512021-05-28T16:06:00+00:00"Bağdatli Yilmaz, Hülya"https://zbmath.org/authors/?q=ai:yilmaz.hulya-bagdatliSummary: The object of the present work is to investigate Sasakian manifolds satisfying certain conditions on \(Q\) tensor whose trace is well-known \(Z\) tensor.A class of strictly convex hypersurfaces satisfying Weingarten-type inequalities. I.https://zbmath.org/1459.530622021-05-28T16:06:00+00:00"Giugiuc, Leonard M."https://zbmath.org/authors/?q=ai:giugiuc.leonard-mihai"Suceavă, Bogdan D."https://zbmath.org/authors/?q=ai:suceava.bogdan-dragosSummary: Linear Weingarten surfaces in three-dimensional ambient space satisfy a relation between mean curvature and Gaussian curvature: \(aH^2+bK=c\). We investigate whether for coreciprocal points on smooth strictly convex hypersurfaces of dimensions 3 and 4 there are any curvature inequalities inspired by the classical Weingarten condition. We also consider the globalization of these pointwise inequalities. This question is suggested by the investigations of Bang-Yen Chen's fundamental inequalities, as we reflect on the geometric interpretations of these relations. The results can be regarded as a response to a question posed by \textit{B.-Y. Chen} in [Proc. Symp. Pure Math. 27, 119--123 (1975; Zbl 0307.53031)]: Let \((M,g)\) be a closed Riemannian manifold. What are the relationships between the total mean curvature of an isometric immersion \(\Phi:M\rightarrow\mathbb{R}^m\) and the Riemannian structure of \((M,g)\)?
For the entire collection see [Zbl 1454.53006].