×

Planar geodesic immersions in pseudo-Euclidean space. (English) Zbl 0572.53042

An isometric immersion \(f: M^ n_ r\to {\mathbb{R}}^ N_ s\) is said to be planar geodesic if the image of each geodesic of M lies in a 2-plane of \({\mathbb{R}}^ N_ s\). Planar geodesic submanifolds of Euclidean space were studied and classified by Hong, Little, Sakamoto, and Ferus; they are spheres and certain Veronese imbeddings of projective spaces.
In the present work, this problem is considered for indefinite metrics. As in the positive-definite case, the length of the second fundamental form is constant, but here the immersion is not necessarily parallel. Among other results, the following list is given of all parallel, planar geodesic, non-totally geodesic, isometric immersions on surfaces in \({\mathbb{R}}^ N_ s:\) the standard imbeddings of pseudo-Riemannian spheres \(S^ 2_ r\) and \(H^ 2_ r\) in \({\mathbb{R}}^ 3_ s\), the Veronese immersions of \(S^ 2_ r\) and \(H^ 2_ r\) in \({\mathbb{R}}^ 5_ s\), and flat surfaces of the form \[ (x,y)\in {\mathbb{R}}^ 2_ r \to (g_ 1(x,y),...,g_ k(x,y),\quad x, y, g_ 1(x,y),...,g_ k(x,y))\in {\mathbb{R}}_{r+k}^{2+2k}, \] with \(k=1,2\), or 3, where the metric on \({\mathbb{R}}_{r+k}^{2+2k}\) has signature (-,..,-,\(+,..,+)\) and \(g_ i\), \(i=1,...,k\), are quadratic polynomials.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Blomstrom, C.: Symmetric immersions in pseudo-Riemannian space forms. Global differential geometry and global analysis 1984. Lect. Notes Math. 1156, 30-45. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0566.53049
[2] Ferus, D.: Symmetric submanifolds of Euclidean space. Math. Ann.247, 81-93 (1980) · Zbl 0446.53041 · doi:10.1007/BF01359868
[3] Hong, S.L.: Isometric immersions of manifolds with plane geodesics into Euclidean space. J. Differ. Geom.8, 259-278 (1973) · Zbl 0282.53018
[4] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. I, II. New York: Interscience 1963, 1969 · Zbl 0119.37502
[5] Little, J.A.: Manifolds with planar geodesics. J. Differ. Geom.11, 265-285 (1976) · Zbl 0319.53038
[6] Magid, M.: Isometric immersions of Lorentz space with parallel second fundamental forms. Tsukuba J. Math.8, 31-54 (1984) · Zbl 0549.53052
[7] Naitoh, H.: Pseudo-Riemannian symmetricR-spaces. Osaka J. Math.21, 733-764 (1984) · Zbl 0556.53031
[8] Sakamoto, K.: Planar geodesic immersions. Tôhoku Math. J.29, 25-56 (1977) · Zbl 0357.53035 · doi:10.2748/tmj/1178240693
[9] Vilms, J.: Submanifolds of Euclidean Space with parallel second fundamental form. Proc. Am. Math. Soc.32, 263-267 (1972) · Zbl 0229.53045
[10] Wolf, J.: Spaces of constant curvature. 3rd ed. Boston: Publish or Perish 1974 · Zbl 0281.53034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.