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A new proof for some optimal inequalities involving generalized normalized \(\delta\)-Casorati curvatures. (English) Zbl 1341.53090

Let \(M\) be an \(n\)-dimensional Riemannian submanifold of a Riemannian manifold \((\bar{M},\bar{g})\). Then it is known that the Casorati curvature of \(M\) is an extrinsic invariant defined as the normalized square of the length of the second fundamental form \(h\) of the submanifold (of dimension \(n\)). This invariant was preferred by Casorati to the traditional Gauss curvature because it corresponds better with the common intuition of curvature. In the present work, analyzing a suitable constrained extremum problem on submanifolds, the authors derive a new proof for two sharp inequalities involving the generalized normalized \(\delta\)-Casorati curvatures of slant submanifolds in quaternionic space forms. These inequalities were previously obtained in [J. W. Lee and G. E. Vîlcu, “Inequalities for generalized normalized \(\delta\)-Casorati curvatures of slant submanifolds in quaternionic space forms”, Taiwan. J. Math. 19, No. 3, 691–702 (2015; doi:10.11650/tjm.19.2015.4832)], using an optimization procedure by showing that a quadratic polynomial in the components of the second fundamental form is parabolic.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
49K35 Optimality conditions for minimax problems
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