Pseudo-Riemannian Submanifolds with 3-planar Geodesics

Year 2017, Volume: 2 Issue: 1, 107 - 112, 01.07.2017

Günay Öztürk Kadri Arslan Betül Bulca

Abstract

In the present paper we study pseudo-Riemannian submanifolds which

have 3-planar geodesic normal sections.We consider W-curves (helices)

on pseudo-Riemannian submanifolds. Finally, we give neccessary and

sufficient condition for a normal section to be a W-curve on pseudo-

Riemannian submanifolds.

Keywords

Pseudo-Riemannian submanifold, geodesic normal sections

3-düzlemsel geodezikli Pseudo-Riemann Altmanifoldlar

Abstract

Bu makalede, 3-düzlemsel normal kesitlere sahip Pseudo-Riemann Altmanifoldları çalıştık.

Pseudo-Riemann Altmanifoldları üzerinde W-eğrilerini(helis) ele aldık. Son olarak, Pseudo-Riemann Altmanifoldları üzerinde

bir normal kesitin W-eğrisi olması için gerek ve yeter koşulu verdik.

Keywords

Pseudo-Riemann Altmanifoldlar, Geodezik normal kesitler

References

• [1] Arslan K. and Celik Y., Submanifolds in real space form with 3-planar geodesic normal sections, Far East J.Math.Sci. 5(1) (1997), 113-120.
• [2] Arslan K., Celik Y. and Deszcz R., A note on geodesic circles on Riemannian manifolds, Far East J.Math.Sci. 5(3) (1997), 453-459.
• [3] Arslan K., Lumiste U., Murathan C. and Ozgur C., 2-Semiparallel surfaces in space forms I,two particular cases, Proc. Estonian Acad.Sci. Phys. Math. 49 (2000), , 139-148.
• [4] Arslan K. and West A.,Product submanifolds with P.3-PNS., Glasgow J. Math. 37 (1995), 73-81.
• [5] Blomstrom C., Planar geodesic immersions in pseudo-Euclidean space, Math. Ann. 274 (1986), 585-598.
• [6] Chen B. Y.,Geometry of submanifolds, Marcel-Dekker, 1973.
• [7] Chen B. Y., Geometry of submanifolds and its applications, Science University of Tokyo 1981.
• [8] Ferus, D., Immersions with parallel second fundamental form, Math. Z 140 (1974), 87-92.
• [9] Ferus D. and Schirrmacher S., Submanifolds in Euclidean space with simple geodesics, Math. Ann. 260 (1982), 57-62.
• [10] Hong S.L., Isometric immersions of manifolds with plane geodesics in Euclidean space, J. Dif. Geo. 8 (1973), 259-278.
• [11] Iawa, T., On some curves in Riemannian geometry, Soochow J. Math. 7 (1980), 37-44.
• [12] Iawa, T., On curves and submanifolds in an indenite-Riemannian manifold, Tukuba J. Math. 9 (1985), 353-371.
• [13] Kim Y.H., Minimal surface of pseudo-Euclidean spaces with geodesic normal sections, Dif. Geo. and its App. 5 (1995), 321-329.
• [14] Kim Y.H., Surfaces in a pseudo-Euclidean space with planar normal sections, J. Geom. 35 (1989), 120-131.
• [15] Li S.J., Isotropic submanifolds with pointwise 3-planar normal sections, Boll. U. M. I. 7 (1987), 373-385.
• [16] Li S.J., Spherical submanifolds with pointwise 3 or 4-planar normal sections, Yokohoma Math. J. 35 (1987), 21-31.
• [17] Little J. A., Manifolds with planar geodesics, J. Diff. Geom. 11 (1976) 265-285.
• [18] Lumiste  U., Submanifolds with Vander Waerden-Bortolotti plane connection and parallelism of the third fundamental form, Izv.Vuzov.Mat. 30(1987), 18-27.
• [19] Murathan C., Pointwise k-planar normal sections with Immersions in warped products of Riemannian manifolds, PhD Thesis, Uludag University, (1995), Bursa-Turkey.
• [20] Nakagawa H., On a certain minimal immersion of a Riemannian manifold into a sphere, Kodai Math. 3 (1980), 321-340.
• [21] Nakanishi Y., On helices and pseudo-Riemannian submanifolds, Tsukuba J. Math. 12 (1988), 459-476.
• [22] Sakamoto K., Helical minimal immersions of compact Riemannian manifolds into a unit sphere, Trans. American Math. Soc. 288 (1985), 765-790.