# An Analytical Analogue of Morse's Lemma

## TL;DR

This paper establishes an analytical analogue of Morse's lemma, demonstrating that near a non-degenerate critical point, the gradient field of a Morse function can be transformed into a standard linear form in suitable local coordinates.

## Contribution

It introduces a method to linearize the gradient field of a Morse function near critical points, extending Morse's topological results to an analytical setting.

## Key findings

- Gradient field near critical points can be expressed in a standard linear form.
- Existence of local coordinates transforming the gradient to a unique linear vector field.
- Constructive proof for transforming Morse gradient fields into standard form.

## Abstract

The Morse function $f$ near a non-degenerate critical point $p$ is understood topologically, in the light of Morse's lemma. However, Morse's lemma standardizes the function $f$ itself, providing little information of how the gradient $\nabla f$ behaves. In this paper, we prove an analytical analogue of Morse's lemma, showing that there exist smooth local coordinates on which a generic Morse gradient field $\nabla f$ near the critical point exhibits a unique linear vector field. We show that on a small neighbourhood of the critical point, the gradient field $\nabla f$ has a natural choice of standard form $V_0(\mathbb{x})=\sum_{i=1}^n \lambda_ix_i\frac{\partial}{\partial x_i}$, and this form only depend on the local behaviour of the Morse function and the Riemannian metric near the critical point. Then we present a constructive proof of the fact that given a generic Morse function $f$, for every critical point, there is a local coordinate on which the gradient field reduces to its standard form.