A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M a€” 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.
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