# Positive definite Hessian on a Riemannian manifold

Consider a Riemannian manifold $$M$$ and a smooth function $$f : M \rightarrow \mathbb{R}$$. Assume that the Hessian of $$f$$, $$\text{Hess}f$$ is positive definite at $$a \in M$$, which is a critical point of $$f$$ i.e. $$\text{grad}f(a) = 0$$.

I wish to understand how to prove the following statement: There exists an open neighbourhood of $$a$$ in $$M$$ where $$f$$ has compact connected sub-level sets all containing $$a$$ and no other critical point of $$f$$. Either a direct answer, sufficiently insightful hints or references would be welcome.

I would request you to supply the definition of the Hessian used while answering the question, since I've seen various versions of the definition, one in which it's a linear map from the tangent space to itself [1], one in which it is a map from the tangent space to the cotangent space[2], one in which it is a bilinear map from the space of vector fields on $$M$$ to $$\mathbb{R}$$ [3].

This effort is aimed at understanding proof of theorem 2 in Mahony et. al.[2].

References

[1] : P. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton, NJ, USA: Princeton University Press, 2008.

[2] : R. Mahony, J. Trumpf, and T. Hamel, “Observers for kinematic systems with symmetry,” IFAC Proceedings Volumes, vol. 46, no. 23, pp. 617 – 633, 27 2013, 9th IFAC Symposium on Nonlinear Control Systems. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1474667016317293

[3] : J. Lafontaine, An Introduction to Differential Manifolds, Springer International Publishing, 2015.

• Hint: Look up the Morse Lemma. – Frieder Jäckel Jul 8 at 21:05
• Thank you, I got the answer to it in John Milnor's Morse Theory. – Anant Joshi Jul 9 at 7:59