open access

https://doi.org/10.14943/83703

https://hdl.handle.net/2115/69307

Umbilicity of spacelike submanifolds of Minkowski space

英語

2002-07

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

558

1-14

We study some properties of spacelike submanifolds in Minkowski n­space all whose points are umbilic with respect to some normal field. As a consequence of these and some results contained in [l] we obtain that being v-umbilic with respect to a parallel lightlike normal field implies conformal flatness for submanifolds of dimension n 􀀩 2 2: 3. In the case of surfaces we relate the umbilicity condition to that of total semiumbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel we show that it is everywhere timelike, spacelike or lightlike if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a 3-dimensional light cone respectively. We also give characterizations of total semiumbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and 4-dimensional light cone.

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