Geometric properties of normal submanifolds
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ORIGINAL ARTICLE
Geometric properties of normal submanifolds Josue´ Mele´ndez1
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Mario Herna´ndez1
Received: 16 August 2019 / Accepted: 9 March 2020 Ó Sociedad Matemática Mexicana 2020
Abstract This paper deals with normal submanifolds immersed in a Riemannian manifold M. We generalized some recent results of surfaces in space forms obtained by Herna´ndez-Lamoneda and Ruiz-Herna´ndez (Bull Braz Marh Soc (NS) 49:447–462, 2018) to arbitrary submanifolds. More precisely, given a submanifold M in M, we study the submanifolds formed by orthogonal geodesics to M, and call it a ruled normal submanifold to M. In the first part of this paper, we analyze these submanifolds and establish some geometric properties of them. Furthermore, we extend some properties about the lines of curvature and using the ideas of [3] also give an extension of the classical Theorem of Bonnet to hypersurfaces of M. Keywords Submanifolds Mean curvature Geodesic Lines of curvature
Mathematics Subject Classification 53B25
1 Introduction The theory of submanifolds is among the most studied fields in differential 3 geometry. The submanifolds in a 3-dimensional space form M are curves and surfaces. Among these, geodesics on a surface and ruled surfaces are some of the 3 most interesting classes of submanifolds in M . There are several classical examples of ruled surfaces in Euclidean space, including the cylinder, cone, helicoid, hyperbolic paraboloid and the hyperboloid of one sheet. One important result in the & Josue´ Mele´ndez [email protected] & Mario Herna´ndez [email protected] 1
Departamento de Matema´ticas, Universidad Auto´noma Metropolitana-Iztapalapa, CP 09340 Ciudad de Me´xico, Me´xico
123
J. Meléndez, M. Hernández
ruled surfaces was given by Catalan [2] in 1842. He proved that the helicoid is the only nonplanar ruled minimal surface in R3 (see also [4, Chapter 3]). Barbosa et al. [1] classified all minimal ruled submanifolds in higher dimensional Riemannian space forms. They showed that locally each minimal ruled submanifold is part of a generalized helicoid. 3 Given a surface M in a space form M , we consider a ruled surface R orthogonal to M along a curve c in M. For instance, for a geodesic c in a surface M R3 , we built a ruled surface R by putting in each point of c the straight line orthogonal to M at the given point. Recently, Herna´ndez-Lamoneda and Ruiz-Herna´ndez [3] proved that R has zero mean curvature along c if and only if c is a geodesic of M. This result relates an intrinsic notion (geodesics) of M with an extrinsic notion (mean curvature) of R. In this paper, we give an extension of this result to submanifolds immersed in a complete oriented Riemannian manifold M (Corollary 3). Moreover, we establish that if N is a totally geodesic submanifold in a hypersurface M immersed in M, then the normal submanifold to M along N, denoted by R, has zero mean curvature along N (see Theorem 2). Roughly speaking, the submanifold R is formed by the geodesics in M normal to M (see Definition 2). Notice that this conc
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