# All zeros of the gradient of a morse function on a riemannian manifold are non-degenerate

Let $$f$$ be a Morse function. Define the gradient vector field on a Riemannian manifold by $$df_p(w)=\langle \text{grad } f|_p, w\rangle \; \text{ for all } p\in M, w\in T_pM.$$ A zero $$p$$ of $$\mathrm{grad}$$ is then a critical point of $$f$$ ($$d_pf = 0$$) and for a local chart $$(U,h)$$ with $$h(p) = 0$$ the Hessian $$\left( \frac{\partial^2(f \circ h^{-1})}{\partial x_i \partial x_j}(0)\right)$$ is nonsingular due to the Morse property of $$f$$.

$$p$$ is a non-degenerate zero of $$\mathrm{grad}$$ if the differential of the pushforward of $$\mathrm{grad}$$ at $$0$$ is an isomorphism. I suppose the latter is just given by the Hessian matrix above but how can I prove it ?

The Hessian of a function can be defined as $$\text{Hess}(f) = \nabla \text{grad} f$$, in other words the Hessian of $$f$$ at a point $$p$$ is the linear map $$T_p M \to T_p M$$ given by $$\text{Hess}(f)_p : v \mapsto \nabla_v(\text{grad}f)|_p.$$ The fact that the function $$f$$ is Morse means the Hessian is nondegenerate and hence at each tangent space this map is an isomorphism. That "the differential of the pushforward of grad at 0 is an isomorphism" should follow by writing the above map in coordinate charts.
• What is $\nabla$ ? – Si Kucing Jul 7 at 2:01
• $\nabla$ is the Levi-Civita connection. Come to think of it, I haven't done Morse theory, so I'm not sure if this is how they define the Hessian there. – Chris Jul 7 at 2:31