Circles and spheres in pseudo-Riemannian geometry

A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index α (0≦α≦n) in a pseudo-Riemannian manifold\(\bar M\) with the metricg and the second fundamental formB. The following theorems are proved. Forε ₀ = +1 or −1,ε ₁ = +1, −1 or 0 (2−2α≦ε ₀+ε ₁≦2n−2α−2) and a positive constantk, every circlec inM withg(c′, c′) = ε ₀ andg(∇ _c′ c′, ∇ _c′ c′) = ε ₁ k ² is a circle in\(\bar M\) iffM is an extrinsic sphere. Forε ₀ = +1 or −1 (−α≦ε₀≦n−α), every geodesicc inM withg(c′, c′) = ε ₀ is a circle in\(\bar M\) iffM is constant isotropic and ∇B(x,x,x) = 0 for anyx ∈ T(M). In this theorem, assume, moreover, that 1≦α≦n−1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When α = 0 orn, this fact does not hold in general.

Authors and Affiliations

1. Department of Mathematics, Faculty of Science, Science University of Tokyo, 162, Tokyo, Japan

   N. Abe, Y. Nakanishi & S. Yamaguchi

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