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

A New Approach to the Fundamental Theorem of Surface Theory

Archive for Rational Mechanics and Analysis, 2007

Let ω be a simply-connected open subset of R 2. Given two smooth enough fields of positive definite symmetric, and symmetric, matrices defined over ω, the well-known fundamental theorem of surface theory asserts that, if these fields satisfy the Gauss and Codazzi-Mainardi relations in ω, then there exists an immersion from ω into R 3 such that these fields are the first and second fundamental forms of the surface (ω) We revisit here this classical result by establishing that a new compatibility relation, shown to be necessary by C. Vallée and D. Fortuné in 1996 through the introduction, following an idea of G. Darboux, of a rotation field on a surface, is also sufficient for the existence of such an immersion. This approach also constitutes a first step toward the analysis of models for nonlinear elastic shells where the rotation field along the middle surface is considered as one of the primary unknowns. Résumé Soit ω un ouvert simplement connexe de R 2. Etant donné deux champs suffisamment réguliers définis dans ω, l'un de matrices symétriques définies positives et l'autre de matrices symétriques, le théorème fondamental de la théorie des surfaces affirme que, si ces deux champs satisfont les relations de Gauss et Codazzi-Mainardi dans ω, alors il existe une immersion de ω dans R 3 telle que ces champs soient les première et deuxième forme fondamentales de la surface (ω). On donne ici une autre approche de ce résultat classique, en montrant qu'une nouvelle relation de compatibilité, dont C. Vallée et D. Fortuné ont montré en 1996 la nécessité en suivant une idée de G. Darboux, estégalement suffisante pour l'existence d'une telle immersion. Cette approche constitueégalement un premier pas vers l'analyse de modèles de coques non linéairement elastiques où le champ de rotations le long de la surface moyenne est pris comme l'une des inconnues principales.

Principles of Differential Geometry

The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on tensor calculus [9, 10] as they are based on the materials and conventions given in those documents. They can be used as a reference for a first course on the subject or as part of a course on tensor calculus.

An Introduction to Riemannian Geometry - Lecture Notes in Mathematics

These lecture notes grew out of an M.Sc. course on differential geometry which I gave at the University of Leeds 1992. Their main purpose is to introduce the beautiful theory of Riemannian Geometry a still very active area of mathematical research. This is a subject with no lack of interesting examples. They are indeed the key to a good understanding of it and will therefore play a major role throughout this work. Of special interest are the classical Lie groups allowing concrete calculations of many of the abstract notions on the menu. The study of Riemannian geometry is rather meaningless without some basic knowledge on Gaussian geometry that i.e. the geometry of curves and surfaces in 3-dimensional space. For this I recommend the excellent textbook: M. P. do Carmo, Differential geometry of curves and surfaces, Prentice Hall (1976). These lecture notes are written for students with a good understanding of linear algebra, real analysis of several variables, the classical theory of ordinary differential equations and some topology. The most important results stated in the text are also proven there. Others are left to the reader as exercises, which follow at the end of each chapter. This format is aimed at students willing to put hard work into the course. For further reading I recommend the interesting textbook: M. P. do Carmo, Riemannian Geometry, Birkhäuser (1992). I am grateful to my many enthusiastic students who throughout the years have contributed to the text by finding numerous typing errors and giving many useful comments on the presentation.

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.