Recent zbMATH articles in MSC 53C35https://zbmath.org/atom/cc/53C352021-05-28T16:06:00+00:00WerkzeugSesquilinear 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].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)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)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].