Ricci curvature of integral submanifolds of an $mathcal{S}$-space form
Bull. Korean Math. Soc. 2007 Vol. 44, No. 3, 395-406
Printed September 1, 2007
Jeong-Sik Kim, Mohit Kumar Dwivedi, and Mukut Mani Tripathi
Chonnam National University, Lucknow University, Lucknow University
Abstract : Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for an integral submanifold of an $\mathcal{S}$-space form. By polarization, we get a basic inequality for Ricci tensor also. Equality cases are also discussed. By giving a very simple proof we show that if an integral submanifold of maximum dimension of an $\mathcal{S}$-space form satisfies the equality case, then it must be minimal. These results are applied to get corresponding results for $C$-totally real submanifolds of a Sasakian space form and for totally real submanifolds of a complex space form.
Keywords : $\mathcal{S}$-space form, integral submanifold, $C$-totally real submanifold, totally real submanifold, Lagrangian submanifold, Ricci curvature, $k$-Ricci curvature, scalar curvature.
MSC numbers : Primary 53C40, 53C15, 53C25
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