×

Improved Chen-Ricci inequality for curvature-like tensors and its applications. (English) Zbl 1230.53053

The author proves several results on a Chen-Ricci inequality involving Ricci curvature and the squared mean curvature for a submanifold \(M\) of ambient manifold \(\widetilde M\).
Contents: Chen-Ricci inequality, improved Chen-Ricci inequality, Lagrangian submanifolds, Kählerian slant submanifolds, \(C\)-totally real submanifolds. Let \(\sigma\) be second fundamental form of the immersion \(M\) into \(\widetilde M\), and \(H\) is mean curvature of \(M\) in \(\widetilde M\). \(\text{Rec}(X)\) denote the Ricci curvature of \(M\) in \(X\in T^1_p M\), and \(T^1_p M\) is the set of unit vectors in tangent space \(T_p M\), \(p\in M\).
The main result is as follows:
Theorem. “Let \(M\) be an \(n\)-dimensional submanifold of a Riemannian manifold. Then, the following statements are true.
(a) For \(X\in T^1_p M\), it follows that \[ \text{Ric}(X)\leq {1\over 4} n^2\| H\|^2+ \widetilde{\text{Ric}}_{(T_p M)}(X),\tag{\(*\)} \] where \(\widetilde{\text{Ric}}_{(T_p M)}(X)\) is the \(n\)-Ricci curvature of \(T_p M\) at \(X\in T^1_p M\) with respect to the ambient manifold \(\widetilde M\).
(b) The equality case of \((*)\) is satisfied by \(X\in T^1_p M\) if and only if \(\sigma(x,y)= \theta\) for all \(y\in T_p M\) orthogonal to \(X\), \(2\sigma(X,X)= nH(p)\). If \(H(p)= 0\), then \(X\in T^1_p M\) satisfies the equality case of \((*)\) if and only if \(X\in N(P)\).
(c) The equality case of \((*)\) holds for all \(X\in T^1_p M\) if and only if either \(p\) is a geodesic point or \(n= 2\) and \(p\) is an umbilical point.”
Here \((*)\) denote the inequality \(| H\|^2\geq {4\over n^2}\{\text{Ric}(X)-(n- 1)c\}\).
The author also studies Chen-Ricci inequality for submanifolds in contact metric manifolds. the exposition is intelligible.

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B25 Local submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Blair, D. E., Geometry of manifolds with structural group \(U(n) \times O(s)\), J. Differential Geom., 4, 155-167 (1970) · Zbl 0202.20903
[2] Blair, D. E., Riemannian geometry of contact and symplectic manifolds, Progr. Math., vol. 203 (2002), Birkhauser Boston, Inc.: Birkhauser Boston, Inc. Boston, MA · Zbl 1011.53001
[3] Bolton, J.; Dillen, F.; Fastenakels, J.; Vrancken, L., A best possible inequality for curvature-like tensor fields, Math. Inequal. Appl., 12, 3, 663-681 (2009) · Zbl 1175.53023
[4] Borrelli, V.; Chen, B.-Y.; Morvan, J.-M., Une caractérisation gómétrique de la sphére de Whitney, C. R. Acad. Sci. Paris Sér. I. Math., 321, 1485-1490 (1995) · Zbl 0842.53016
[5] Cartan, É., Lecons sur la géométrie des espaces de Riemann (1946), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0060.38101
[6] Castro, I.; Urbano, F., Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form, Tôhoku Math. J., 45, 582-656 (1993) · Zbl 0792.53050
[7] Castro, I.; Urbano, F., Twistor holomorphic Lagrangian surface in the complex projective and hyperbolic planes, Ann. Global Anal. Geom., 13, 59-67 (1995) · Zbl 0827.53049
[8] Chen, B.-Y., Geometry of Slant Submanifolds (1990), K.U. Leuven: K.U. Leuven Leuven · Zbl 0716.53006
[9] Chen, B.-Y., Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel), 60, 6, 568-578 (1993) · Zbl 0811.53060
[10] Chen, B.-Y., Jacobiʼs elliptic functions and Lagrangian immersions, Proc. Roy. Soc. Edinburgh, 126, 687-704 (1996) · Zbl 0866.53045
[11] Chen, B.-Y., Strings of Riemannian invariants, inequalities, ideal immersions and their applications, (The Third Pacific Rim Geometry Conference. The Third Pacific Rim Geometry Conference, Seoul, 1996. The Third Pacific Rim Geometry Conference. The Third Pacific Rim Geometry Conference, Seoul, 1996, Geom. Topol. Monogr., vol. 25 (1998), Int. Press: Int. Press Cambridge, MA), 7-60 · Zbl 1009.53041
[12] Chen, B.-Y., Interaction of Legendre curves and Lagrangian submanifolds, Israel J. Math., 99, 69-108 (1997) · Zbl 0884.53014
[13] Chen, B.-Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasg. Math. J., 41, 33-41 (1999) · Zbl 0962.53015
[14] Chen, B.-Y., On slant surfaces, Taiwanese J. Math., 3, 2, 163-179 (1999) · Zbl 1013.53038
[15] Chen, B.-Y., On Ricci curvature of isotropic and Langrangian submanifolds in complex space forms, Arch. Math. (Basel), 74, 154-160 (2000) · Zbl 1037.53041
[16] Chen, B.-Y., Riemannian submanifolds, (Dillen, F.; Verstraelen, L., Handbook of Differential Geometry, vol. I (2000), North-Holland: North-Holland Amsterdam), 187-418 · Zbl 0968.53002
[17] Chen, B.-Y., Riemannian geometry of Lagrangian submanifolds, Taiwanese J. Math., 5, 4, 681-723 (2001) · Zbl 1002.53053
[18] Chen, B.-Y., Classification of slumbilical submanifolds in complex space forms, Osaka J. Math., 39, 23-47 (2002) · Zbl 1002.53039
[19] Chen, B.-Y., \(δ\)-invariants, inequalities of submanifolds and their applications, (Topics in Differential Geometry (2008), Ed. Acad. Române: Ed. Acad. Române Bucharest), 29-155 · Zbl 1181.53001
[20] Chen, B.-Y.; Dillen, F.; Verstraelen, L., \(δ\)-invariants and their applications to centroaffine geometry, Differential Geom. Appl., 22, 341-354 (2005) · Zbl 1075.53012
[21] Chen, B.-Y.; Ogiue, K., Two theorems on Kaehler manifolds, Michigan Math. J., 21, 225-229 (1974) · Zbl 0295.53028
[22] Chen, B.-Y.; Vrancken, L., Lagrangian submanifolds satisfying a basic inequality, Math. Proc. Cambridge Philos. Soc., 120, 291-307 (1996) · Zbl 0860.53034
[23] Deng, S., An improved Chen-Ricci inequality, Int. Electron. J. Geom., 2, 2, 39-45 (2009) · Zbl 1191.53037
[24] Hong, S. P.; Matsumoto, K.; Tripathi, M. M., Certain basic inequalities for submanifolds of locally conformal Kaehler space forms, SUT J. Math., 41, 1, 75-94 (2005) · Zbl 1091.53033
[25] Hong, S. P.; Tripathi, M. M., On Ricci curvature of submanifolds, Int. J. Pure Appl. Math. Sci., 2, 2, 227-245 (2005) · Zbl 1131.53309
[26] Hong, S. P.; Tripathi, M. M., On Ricci curvature of submanifolds of generalized Sasakian space forms, Int. J. Pure Appl. Math. Sci., 2, 2, 173-201 (2005) · Zbl 1131.53308
[27] Hong, S. P.; Tripathi, M. M., Ricci curvature of submanifolds of a Sasakian space form, Iran. J. Math. Sci. Inform., 1, 2, 31-52 (2006) · Zbl 1301.53051
[28] Kim, J.-S.; Dwivedi, M. K.; Tripathi, M. M., Ricci curvature of integral submanifolds of an \(S\)-space form, Bull. Korean Math. Soc., 44, 3, 395-406 (2007) · Zbl 1142.53043
[29] Liu, X., On Ricci curvature of totally real submanifolds in a quaternion projective space, Arch. Math. (Brno), 38, 297-305 (2002) · Zbl 1090.53052
[30] Matsumoto, K.; Mihai, I.; Tazawa, Y., Ricci tensor of slant submanifolds in complex space forms, Kodai Math. J., 26, 1, 85-94 (2003) · Zbl 1049.53042
[31] Mihai, I., Ricci curvature of submanifolds in Sasakian space forms, J. Aust. Math. Soc., 72, 2, 247-256 (2002) · Zbl 1017.53052
[32] Mihai, I., On Kaehlerian slant submanifolds in complex space forms satisfying a geometrical inequality, An. Univ. Bucuresti Mat., 53, 1, 77-84 (2004) · Zbl 1112.53041
[33] Oprea, T., On a geometric inequality (3 November 2005)
[34] Oprea, T., Ricci curvature of Lagrangian submanifolds in complex space forms, Math. Inequal. Appl., 13, 4, 851-858 (2010) · Zbl 1210.53074
[35] Sahin, B., Every totally umbilical proper slant submanifold of a Kähler manifold is totally geodesic, Results Math., 54, 167-172 (2009) · Zbl 1178.53056
[36] Tanno, S., Sasakian manifolds with constant \(φ\)-holomorphic sectional curvature, Tôhoku Math. J. (2), 21, 501-507 (1969) · Zbl 0188.26801
[37] Tripathi, M. M., Chen-Ricci inequality for submanifolds of contact metric manifolds, J. Adv. Math. Stud., 1, 1-2, 111-134 (2008) · Zbl 1181.53048
[38] Wu, H., Manifolds of partially positive curvature, Indiana Univ. Math. J., 36, 3, 525-548 (1987) · Zbl 0639.53050
[39] Yamaguchi, S.; Kon, M.; Ikawa, T., \(C\)-totally real submanifolds, J. Differential Geom., 11, 1, 59-64 (1976) · Zbl 0329.53039
[40] Yano, K.; Kon, M., Anti-Invariant Submanifolds, Lect. Notes Pure Appl. Math., vol. 21 (1976), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York, Basel · Zbl 0349.53055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.