Melissa Sugimoto
Applications of the Inverse Function Theorem in Differential Geometry
Overview: We have studied some facts about differential geometry, a field of mathematics which has many applications in topics such as physics, biology, and computer graphics. In this presentation, we will present a proof and some mathematical applications of a very important theorem, the Inverse Function Theorem.Abstract: We have studied some facts about differential geometry, a field of mathematics which has many applications in topics such as physics, biology, and computer graphics. In this presentation, we will discuss the Inverse Function Theorem. The Inverse Function Theorem is a critical theorem. Versions of it appear across differential calculus and geometry, and it can even be generalized to abstract manifolds in higher dimensions. The idea of the theorem, as given in [1] is that the derivative of a map is locally such a good approximation of the map that its invertibility propagates to the invertibility of the map itself. We present a proof of the Inverse Function Theorem and demonstrate some applications that allow us to more easily describe the nature of surfaces embedded in three-dimensional space. We then generalize the setting to manifolds of higher dimensions and show that the Inverse Function Theorem can be applied in the same way to prove an analogous result, highlighting its power as a tool in differential geometry and topology.
[1] Nicolaescu, Liviu. Lectures on the Geometry of Manifolds. 31 August 2009.
Zoom Link