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Some theorems on totally geodesic mappings. (English) Zbl 0628.53054

The author deals with relative affine and harmonic maps and a minimal immersion from a complete Riemannian manifold M to a Riemannian manifold N. He investigates conditions under which those maps become totally geodesic. The main tools are formulas concerning the Laplacian of the square of the length of the second fundamental form and Omori’s maximum principle. One of the main results is stated as follows: suppose that \(f: M\to N\) is a minimal immersion from a complete Riemannian manifold M to a Riemannian manifold n with nonnegative constant sectional curvature C. If the sectional curvature of \(M\geq c/2\), then f is a totally geodesic map.
Reviewer: T.Ishihara

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
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