Non-diagonal metric on a product riemanniann manifold

TL;DR

This paper introduces a new class of metrics called generalized warped products on product manifolds, analyzes their curvature, and explores the relationships between the geometries of base, fiber, and total space.

Contribution

It constructs symmetric tensor fields that generalize warped products, derives curvature expressions, and computes the Laplacian-Beltrami operator for these metrics.

Findings

Conditions for $G_{f_1f_2}$ to be a metric tensor

Explicit curvature formulas for generalized warped products

Relationships between geometries of base, fiber, and total space

Abstract

In this paper, We construct the symmetric tensor field $G_{f_1f_2}$ and $h_{f_1f_2}$ on a product manifold and we give conditions under which $G_{f_1f_2}$ becomes a metric tensor, theses tensors fields will be called the generalized warped product, and then we develop an expression of curvature for the connection of the generalized warped product in relation to those corresponding analogues of its base and fiber and warping functions. By constructing a frame field in $M_1\times_{f_1f_2}M_2$ with respect to the Riemannian metric $G_{f_1f_2}$ and $h_{f_1f_2}$, then we calculate the Laplacian$-$Beltrami operator of a function on a generalized warped product which may be expressed in terms of the local restrictions of the functions to the base and fiber. Finally, we conclude some interesting relationships between the geometry of the couples $(M_1,g_1)$ and $(M_2,g_2)$ and that of $(M_1\times M_2,h_{f_1f_2})$.