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Ricci tensor of slant submanifolds in a quaternion projective space. (English) Zbl 1223.53043

The authors prove the following: Suppose that \(M\) is an \(n\)-dimensional \(\theta\)-slant submanifold in a quaternion projective space \(\mathbb QP^m (4c)\), then, for each unit vector \(X\in T_p M,\) the Ricci tensor \(S(X)\) satisfies the inequality
\[ S(X)\leq \tfrac{n^2}{4}\,H^2 +(n-1)c + 9c \cos ^2\theta. \]
Equality holds identically for all unit tangent vectors at \(p\) if and only if \(p\) is a totally geodesic point or \(n=2\) and \(M\) is a totally umbilical point.

MSC:

53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:

[1] Chen, B. Y., Geometry of Slant Submanifolds (1990), K.U. Leuven · Zbl 0716.53006
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