Foundations of Morse Theory and Applications

The Morse theory is a sophisticated approach to explore the shape of a manifold by studying certain of its real-valued functions which can be expressed as non-degenerate canonical quadratic forms in some local coordinate systems about their critical point. In this project, we focus on the basic theorems of Morse Theory including Reeb's result which states that every compact finite dimensional smooth manifold that admits a smooth function with exactly two critical points which are both non-degenerate, is homeomorphic to a sphere.