#### Title

#### Date of Award

6-2010

#### Document Type

Union College Only

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

Christina Tonnesen-Friedman

#### Language

English

#### Keywords

geometry; vector, differential; structure, manifold; metric

#### Abstract

This paper will be a brief introduction to the theories of diﬀerential geometry. The foundation of this paper will be based on the concept of a manifold. For most of our mathematical careers, we have been restricted to the Euclidean space as our model of a geometric space. The Euclidean space is very nice in that we can deﬁne diﬀerentiability, functions, mappings, tangents spaces, vector ﬁelds, and even a metric to transform the Euclidean space into a metric space. Yet the Euclidean space is an example of a spe-ciﬁc geometric space and the theory of diﬀerential geometry will examine geometric structures from a general perspective. The foundation of our theory will begin with the introduction of a man-ifold. The manifold will be our abstraction of a geometric structure and from here we will develop the same ideas on a manifold that are present in Euclidean space. We will ﬁrst introduce the idea of the concept of a topological manifold, then proceed to deﬁne a diﬀerentiable structure on the manifold. The crux of this paper will involve this central idea of deﬁning a diﬀerentiable structure on a manifold and providing key example and proofs of certain kinds of manifolds being diﬀerentiable manifolds. We will then continue our discussion by deﬁning submanifolds and properties of subman-ifolds and introduce a group structure on certain manifolds that transforms a diﬀerentiable manifold into a Lie Group. The preliminary concept of diﬀerentiability on a manifold will lead us into a discussion of tangent spaces at points on a manifold. Then naturally we will deﬁne the concept of a vector ﬁeld on a manifold. The central motivation for deﬁning these structures on a manifold will eventually be to deﬁne what is called a Riemannian metric on a manifold. This will allow us to transform the tangent space at any point in the manifold into a euclidean space. We will end our discussion with a brief introduction of curvature and the Einstein manifold which relates the curvature with the Riemannian metric.

#### Recommended Citation

Kabir, Foyroj, "Differentiable manifolds" (2010). *Honors Theses*. 1158.

https://digitalworks.union.edu/theses/1158