Another approach to the fundamental theorem of Riemannian geometry in , by way of rotation fields (original) (raw)

Riemannian Geometry and Modern Developments

GANIT: Journal of Bangladesh Mathematical Society, 2019

In this paper, we compute the Christoffel Symbols of the first kind, Christoffel Symbols of the second kind, Geodesics, Riemann Christoffel tensor, Ricci tensor and Scalar curvature from a metric which plays a fundamental role in the Riemannian geometry and modern differential geometry, where we consider MATLAB as a software tool for this implementation method. Also we have shown that, locally, any Riemannian 3-dimensional metric can be deformed along a directioninto another metricthat is conformal to a metric of constant curvature GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 71-85

Uniqueness theorem of the curvature tensor

This paper develops the uniqueness theorem of the curvature tensor, which states that the Riemann-Christoffel tensor (and its linear combinations) is the only tensor that depends on the connection and is linear with respect to the second derivatives of the metric tensor. From this result, Cartan's theorem is obtained, according to which Einstein's tensor is the only second-order tensor that depends on the metric tensor, on its first derivatives, is linear with respect to the second derivatives of the metric tensor and its covariant divergence is null, admitting that the coefficients of these second derivatives are tensors derived from the metric tensor.

On rotationally invariant vector fields in the plane

Manuscripta Mathematica, 1996

This work is concerned with global properties of a class of C-valued vector fields in the plane which are rotationally invariant. It is shown that the finite type rotationally invaxiant vector fields have global first integrals. We also study the global hypoellipticity and global solvability properties of these vector fields. Note that L is orthogonal to w in the sense of the usual pairing between 1-forms and vector fields.

A Stokes theorem for second-order tensor fields and its implications in continuum mechanics

International Journal of Non-Linear Mechanics, 2005

We give a constructive proof of a particular Stokes theorem (1.4) for tensor fields in R 3 ⊗ R 3 . Its specialization to symmetric tensor fields, given in (1.5), bears a close relation to compatibility in linear elasticity theory and to the generalized Beltrami representation of symmetric tensor fields in continuum mechanics. These issues are discussed.

Some Applications of Integral Formulas in Riemannian Geometry and PDE?s

Milan Journal of Mathematics, 2003

We show some applications of integral formulas, most notably the Hsiung-Minkowski formulas, the Rellich-Pohozaev identities, and variations thereof, to the study of geometric problems and PDE's. Our presentation aims at underlining the common geometrical and analytical features of such formulas. 220 S. Pigola, M. Rigoli, and A.G. Setti Vol. 71 (2003) This can be simply understood with the following two examples. Let (M, g) be compact and m-dimensional and let λ 1 denote the first (nonzero) eigenvalue of the problem