Riemannian Warped Product Submersions
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Riemannian Warped Product Submersions ˙ Irem K¨ upeli Erken and Cengizhan Murathan Abstract. In this paper, we introduce Riemannian warped product submersions and construct examples and give fundamental geometric properties of such submersions. On the other hand, a necessary and sufficient condition for a Riemannian warped product submersion to be totally geodesic, totally umbilic and minimal is given. Mathematics Subject Classification. Primary 53B20, 53C17, 53C42, 53C55. Keywords. Riemannian immersion, Riemannian submersion, warped product.
1. Introduction Warped product manifolds play very important roles to construct cosmological models in general relativity theory. For instance, Schwarzschild and RobertsonWalker cosmological models are well known examples of warped product manifolds [20]. It is well known that the notion of warped product manifolds appeared in the differential geometry as a generalization of the Riemannian product manifolds [2]. Historically, such spaces have been used in order to prove that some classes of Riemannian manifolds are not empty and to produce large families of examples. On the other hand, one of the methods used to classify mathematical objects is to split into mathematical elements. For example, the theorem of de Rham characterizies Riemannian manifolds that can be locally decomposed as a Riemannian product manifold. Followed by this theorem, the J. Moore’s Theorem gives sufficient conditions for an isometric immersion into a Euclidean space to decompose into a product immersion [14]. A warped product immersion is defined as follows: Let M0 ×ρ1 M1 × · · · ×ρk Mk be a warped product and let fi : Ni → Mi , i = 0, . . . , k, be 0123456789().: V,-vol
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I. K. Erken and C. Murathan
Results Math
isometric immersions, and define σi := ρi ◦ f0 : N0 → R+ for i = 1, . . . , k. Then the map f : N0 ×ρ1 N1 × · · · ×ρk Nk → M0 ×ρ1 M1 × · · · ×ρk Mk given by f (x0 , . . . , xk ) := (f0 (x0 ), f1 (x1 ), . . . , fk (xk )) is an isometric immersion, which is called a warped product immersion [8,18]. Nash embeding theorem which was given by J. F. Nash state that every Riemann manifold can be isometrically immersed in some Euclidean spaces with sufficently high dimensions [17]. Due to the Nash’s theorem, one can say that every warped product M0 ×ρ1 M1 manifold can be embedded to some Euclidean spaces. In view of Nash’s theorem, the following decomposition theorem of S. N¨ o lker is known as a generalization of J. Moore’s Theorem [14]. Theorem 1. [18] Let φ : N0 ×ρ1 N1 × · · · ×ρk Nk → RN (c) be an isometric immersion into a Riemannian manifold of constant curvature c. If h is the second fundamental form of φ and h(Xi , Xj ) = 0, for all vector fields Xi and Xj , tangent to Ni and Nj respectively, with i = j, then, locally, φ is a warped product immersion. Recently B. Y. Chen studied fundamental geometric properties of warped product immersions and collected these results in extensive and comprehensive survey of warped product manifolds and submanifolds [7]. It is a
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