Abstract
In the present paper, we prove that if Laplacian for the warping function of complete warped product submanifold M m = B p × h F q in a unit sphere S m + k satisfies some extrinsic inequalities depending on the dimensions of the base B p and fiber F q such that the base B p is minimal, then M m must be diffeomorphic to a unit sphere S m . Moreover, we give some geometrical classification in terms of Euler–Lagrange equation and Hamiltonian of the warped function. We also discuss some related results.
Highlights
Introduction and Main ResultsWe will use the following acronyms throughout the paper: ‘warped product (WP)’ for Warped product, ‘WF’ for warping function, ‘Riemannian manifolds (RMs)’ for Riemannian manifold, and ‘SFF’ for second fundamental form
If (B, gB ) and (F, gF ) are two Riemannian manifolds (RMs), and h is a positive differentiable function defined on the base manifold B, we define the metric g = π ∗ gB + h2 σ∗ gF on the product manifold B × F, where π and σ are the projection maps on
The main goal of this note is to extend the rigidity Theorem 1 to a complete warped product submanifolds and find the solution for our proposed problem where motivation comes from the Nash embedding theorem [25] which states that “every Riemannian manifold has an isometric immersion into Euclidean space of sufficient high codimension”
Summary
We will use the following acronyms throughout the paper: ‘WP’ for Warped product, ‘WF’ for warping function, ‘RM’ for Riemannian manifold, and ‘SFF’ for second fundamental form. The non-existence of a compact stable minimal submanifold or stable currents is sharply associated with the topology and geometric function theory on Riemannian structure of the whole manifold It has been shown in [14] that if the sectional curvature of a compact oriented minimal submanifold M of dimension m in the unit sphere Sm+k with codimension p satisfies some pinching p.sign( p−1). What is the best pinching constant for the differentiable rigidity sphere theorem of complete minimal warped product submanifold in a unite sphere under pinching conditions using the Laplace operator for the warping function?. The main goal of this note is to extend the rigidity Theorem 1 to a complete warped product submanifolds and find the solution for our proposed problem where motivation comes from the Nash embedding theorem [25] which states that “every Riemannian manifold has an isometric immersion into Euclidean space of sufficient high codimension”. We noticed that Theorems 2 and 3 are differentiable sphere theorems for complete warped product submanifolds without assumption that Mn is connected
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