Isometric immersions of Riemannian manifolds - Project Euclid
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Isometric immersions of Riemannian manifolds - Project Euclid
210. H. OMORI submanifold of $M$. Let $m=\min\{H_{q}(X, X);q\in\tilde{S}_{p}(a), \Vert X\Vert=1\}$ . Then,. $(\nabla\nabla f)(q_{k})(X, ...
J. Math. Soc. Japan Vol. 19, No. 2, 1967
Isometric immersions of Riemannian manifolds By Hideki OMORI (Received Oct. 24, 1966)
Introduction. It seems to be natural in differential geometry to conjecture that many theorems proved under the condition that the underlying manifolds are compact can be extended to manifolds with complete Riemannian metric under some suitable additional conditions. Now, it is clear that for any smooth function on a compact manifold there is a point such that $\nabla_{i}f(p)=0$ and is negative semi-definite. It is also clear that if a smooth function $f(x)$ on a real line has an upper bound, then for any . and are there is $x\in R$ such that This simple fact, however, can not be extended in general to a complete Riemannian manifold. That is, there is a complete Riemannian manifold $M$ and a bounded function on $M$ such that $m(p)=\{X^{i}X^{j}\nabla_{i}\nabla_{j}f(p);\Vert X\Vert=1\}$ is always larger than $a>0$ . This example can easily be constructed on with metric $f$
