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If there is a function $f : M \to \mathbb R$ then the critical point is given as a point where

$$d f = 0$$

$df$ being 1-form (btw am I right here?). Is there a coordinate independent formulation of a criteria to determine if this point is a local maximum or minimum (or a saddle point)?

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There are coordinate free definitions of second derivatives. Positive/negative definiteness of this object is a coordinate free criterion. –  user20266 Jul 18 '12 at 16:11

2 Answers 2

Let $p$ be a critical point for a smooth function $f:M\to \mathbb{R}.$
Let $(x_\,\ldots,x_n)$ be an arbitrary smooth coordinate chart around $p$ on $M.$
From multivariate calculus we know that a sufficient condition for $p$ to be a local maximum (resp. minimum) of $f$ is the positiveness (resp. negativeness) of the Hessian $H(f,p)$ of $f$ at $p$ which is the bilinear map on $T_pM$ defined locally by $$H(f,p)=\left.\frac{\partial^2f}{\partial x_i\partial x_j}dx^i\otimes dx^j\right|_p,$$ here the Einstein convention on summation is working.

However, as Thomas commented, the Hessian of a function at a critical point has a coordinate-free espression.
Infact, $H(f,p): T_pM\times T_pM\to\mathbb{R}$ is characterized by $$H(f,p)(X(p),Y(p))=(\left.\mathcal{L}_X(\mathcal{L}_Y f))\right|_p$$ for any smooth vector fields $X$ and $Y$ on $M$ around $p.$


Note that without a Riemannian metric on $M$ you cannot invariantly define the Hessian of a function at a non-critical point.

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Dear Giuseppe, I have upvoted you but I certainly don't believe that the hessian is an alternating $2$-form as in the displayed equality you wrote: it is a symmetric form (or, equivalently, a quadratic form). –  Georges Elencwajg Jul 18 '12 at 21:17
    
and we know the signature of a quadratic form is co-ordinate independent by Sylvester's law of inertia. –  Kris Jul 18 '12 at 21:39

The generalization of the Hessian matrix to functions on smooth manifolds is

$$H(X,Y) = X (Yf) - df (\nabla_X Y)$$

where $X$ and $Y$ are vector fields, i.e. the Hessian is a bilinear form. The definition for positive/negative definiteness for bilinear forms is the usual one. $H$ is positive definite if

$$H(X,X)>0$$

for all vector fields $X$, and similarly for negative definite. As usual, a critical point is a local maximum if $H$ is negative definite, a local minimum if $H$ is positive definite, and a saddle otherwise.

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