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.