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Hypersurfaces whose tangent geodesics omit a nonempty set. (English) Zbl 0723.53036
Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 1-13 (1991).

[For the entire collection see Zbl 0718.00010.]

Let M n be an n-dimensional connected, complete manifold and x: M n Q c n+1 be an immersion, where Q c n+1 is a space form of constant curvature c. For every pM n , let (Q c n ) p be the totally geodesic hypersurface of Q c n+1 tangent to x(M n ) at x(p). We set W=Q c n+1 - pM (Q c n ) p · In this interesting paper the authors study the immersions for which the set W is nonempty. In the case c=0, a result of B. Halpern [Proc. Am. Math. Soc. 30, 181-184 (1971; Zbl 0222.57016)] states that, for a compact n-dimensional (n2) manifold immersed in R n+1 , the set W is nonempty if and only if M n is embedded as the boundary of an open starshaped set. In the present paper, after some preliminaries concerning the position vector and support function, the authors show that the same happens when the ambient space is Q c n+1 , c arbitrary. Moreover, the authors prove the following: Let M n be a complete Riemannian manifold and let x: M n Q c n+1 be an isometric minimal immersion. If the set W is open and nonempty, then x is totally geodesic. This result for minimal immersions x: M 2 Q c 3 , c0, with nonempty W has been proved by the reviewer and D. Koutroufiotis [Trans. Am. Math. Soc. 281, 833-843 (1984; Zbl 0538.53057)].

53C42Immersions (differential geometry)
53C40Global submanifolds (differential geometry)