# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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: ${M}^{n}\to {Q}_{c}^{n+1}$ be an immersion, where ${Q}_{c}^{n+1}$ is a space form of constant curvature c. For every $p\in {M}^{n}$, let ${\left({Q}_{c}^{n}\right)}_{p}$ be the totally geodesic hypersurface of ${Q}_{c}^{n+1}$ tangent to $x\left({M}^{n}\right)$ at x(p). We set $W={Q}_{c}^{n+1}-{\cup }_{p\in M}{\left({Q}_{c}^{n}\right)}_{p}·$ 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 (n$\ge 2\right)$ 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: ${M}^{n}\to {Q}_{c}^{n+1}$ be an isometric minimal immersion. If the set W is open and nonempty, then x is totally geodesic. This result for minimal immersions x: ${M}^{2}\to {Q}_{c}^{3}$, $c\ge 0$, with nonempty W has been proved by the reviewer and D. Koutroufiotis [Trans. Am. Math. Soc. 281, 833-843 (1984; Zbl 0538.53057)].

##### MSC:
 53C42 Immersions (differential geometry) 53C40 Global submanifolds (differential geometry)