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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.

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    $\begingroup$ Hint: Look up the Morse Lemma. $\endgroup$ Jul 8, 2019 at 21:05
  • $\begingroup$ Thank you, I got the answer to it in John Milnor's Morse Theory. $\endgroup$ Jul 9, 2019 at 7:59

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Found the answer in John Milnor's Morse Theory [1], Lemma 2.2.

I also found a set of lecture notes particularly useful in understanding this concept [2].

References

[1] : John Milnor, Morse Theory, Princeton University Press, 1963.

[2] : An Introductory Treatment Of Morse Theory On Manifolds, Amy Hua. Available: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Hua.pdf.

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