I want to analyze the local behavior of a function $f(\mathbf{x})$ on a manifold at a certain point $\mathbf{x}_0$ which lies on the manifold.
One approach would be to add small disturbances $\mathbf{h}$ to $\mathbf{x}_0$, project $\mathbf{x} = \mathbf{x}_0 + \mathbf{h}$ onto the manifold (closest point $\mathbf{x}'$ on the manifold), and check whether e.g. $\Delta f = f(\mathbf{x}') - f(\mathbf{x}_0) < 0$ for all $\mathbf{h}$ which would indicate a local maximum.
However, it may be difficult to analytically find the projection $\mathbf{x}'$ on the manifold. Therefore I thought of projecting $\mathbf{x}$ onto the tangent space of the manifold at $\mathbf{x}_0$ instead.
My question is: If I can show that $f$ e.g. has a local maximum at $\mathbf{x}_0$ on the tangent space, can I conclude that it also has a local maximum on the manifold?
(Edit: There seems to be an additional difficulty besides the question whether locally the (smooth) manifold can be represented by its tangent space for this analysis: If we derive an expression for $\Delta f(\mathbf{h})$ e.g. at a local maximum, we will always see $\Delta f(\mathbf{h}) = 0$ at the direction normal to the manifold at $\mathbf{x}_0$, i.e. for $\mathbf{h}' = \mathbf{0}$. I wonder whether this can be eliminated afterwards.)