caa a22 4500 465825109 CHVBK 20180323112147.0 cr unu---uuuuu 170327e19900401xx s 000 0 eng 10.1007/BF01833144 doi (NATIONALLICENCE)springer-10.1007/BF01833144 Circles and spheres in pseudo-Riemannian geometry [Elektronische Daten] [N. Abe, Y. Nakanishi, S. Yamaguchi] 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. Birkhäuser Verlag, 1990 Primary 53C50, 53C40, 53B30, 53B25 nationallicence Abe N. Department of Mathematics, Faculty of Science, Science University of Tokyo, 162, Tokyo, Japan aut Nakanishi Y. Department of Mathematics, Faculty of Science, Science University of Tokyo, 162, Tokyo, Japan aut Yamaguchi S. Department of Mathematics, Faculty of Science, Science University of Tokyo, 162, Tokyo, Japan aut aequationes mathematicae Birkhäuser-Verlag 39/2-3(1990-04-01), 134-145 0001-9054 39:2-3<134 1990 39 10 https://doi.org/10.1007/BF01833144 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01833144 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Abe N. Department of Mathematics, Faculty of Science, Science University of Tokyo, 162, Tokyo, Japan aut NATIONALLICENCE 700 1- Nakanishi Y. Department of Mathematics, Faculty of Science, Science University of Tokyo, 162, Tokyo, Japan aut NATIONALLICENCE 700 1- Yamaguchi S. Department of Mathematics, Faculty of Science, Science University of Tokyo, 162, Tokyo, Japan aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 39/2-3(1990-04-01), 134-145 0001-9054 39:2-3<134 1990 39 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer