Summary
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.
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Abe, N., Nakanishi, Y. & Yamaguchi, S. Circles and spheres in pseudo-Riemannian geometry. Aeq. Math. 39, 134–145 (1990). https://doi.org/10.1007/BF01833144
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DOI: https://doi.org/10.1007/BF01833144