Date of Award


Document Type

Union College Only

Degree Name

Bachelor of Science



First Advisor

Christina Tonnesen-Friedman




geometry; vector, differential; structure, manifold; metric


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.