zbMATH — the first resource for mathematics

On a Riemannian invariant of Chen type. (English) Zbl 1195.53072
In [An. Univ. Bucur., Mat. 54, No. 1, 127–136 (2005; Zbl 1150.53340)] we proved Chen’s inequality regarded as a problem of constrained maximum. In this paper we introduce a Riemannian invariant obtained from Chen’s invariant, replacing the sectional curvature by the Ricci curvature of \(k\)-order. This invariant can be estimated, in the case of submanifolds \(M\) in space forms \(\widetilde{M}(c)\), varying with \(c\) and the mean curvature of \(M\) in \(\widetilde{M}(c)\).

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Zbl 1150.53340
Full Text: DOI arXiv
[1] B.Y. Chen, Some pinching classification theorems for minimal submanifolds , Arch. Math. 60 (1993), 568-578. · Zbl 0811.53060
[2] ——–, A Riemannian invariant for submanifolds in space forms its applications , in Geometric topology of submanifolds , World Scientific, Leuven, Brussel, · Zbl 0865.53050
[3] ——–, A Riemannian invariant its applications to submanifolds theory , Results Math. 27 (1995), 17-26. · Zbl 0834.53045
[4] ——–, Mean curvature and shape operator of isometric immersions in real-space-forms , Glasgow Math. J. 38 (1996), 87-97. · Zbl 0866.53038
[5] ——–, Some new obstructions to minimal Lagrangian isometric immersions , Japan. J. Math. 26 (2000), 105-127. · Zbl 1026.53009
[6] T. Oprea, Optimizations on Riemannian submanifolds , An. Univ. Bucharest 54 (2005), 127-136. · Zbl 1150.53340
[7] C. Udrişte, Convex functions and optimization methods on Riemannian manifolds , in Mathematics and its applications , 297 , Kluwer Academic Publishers Group, Dordrecht, 1994. · Zbl 0932.53003
[8] C. Udrişte, O. Dogaru and I \cTevy, Extrema with nonholonomic constraints , Geometry Balkan Press, Bucharest, 2002.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.