This is from an engineering assignment on Riemannian manifolds and I confess that this is my first experience with manifolds. Here is the problem:
Let $\mathcal{M}\subset \mathbb{R}^2$ be an ellipse of fixed major and minor axis. Lets say the embedding is $x=a\cos(\theta)$ and $y=b \sin(\theta)$. Let $f(\theta): \mathcal{M} \rightarrow \mathbb{R}$ be a smooth function on the ellipse. I wish to know what the gradient and Riemannian Hessian of $f$ will be (using local coordinates).
My attempt:
The Jacobian is $D=[-a \sin(\theta), b \cos(\theta)]^T$ so the inherited metric is $g_{1,1}=D^tD=a^2\sin^2(\theta)+b^2 \cos^2(\theta)$. Now the gradient should be $\Delta f= g^{1,1}\frac{d}{d\theta}=\left(a^2\sin^2(\theta)+b^2 \cos(\theta)\right)^{-1} \frac{df}{d\theta}$. Here the superscript is for inverse.
For the Riemannian Hessian I think it should just be $\frac{d^2}{d\theta^2}$ but I do not know how to compute it.
My intuition in this problem is that it should coincide with the result from change of coordinates: Writing the gradient like $\left(\frac{df}{dx}, \frac{df}{dy}\right)$ and then using the chain rule. Similarly for the Hessian matrix $H_{i,j=1}^2=\frac{d^2f}{dx^idx^j}$ where $x^1=x, x^2=y$. However, the answers I got do not coincide with that. If it helps solve the problem, note that eventually I am hoping to verify that the trace of the Hessian is the Laplace-Beltrami, which I have also not calculated yet but hope that I can after this is figured out.
Best Answer
Your calculation of the gradient is correct. The Hessian can be calculated using the formula for the Hessian
$$ Hf\left( \frac{\partial}{\partial \theta}, \frac{\partial}{\partial \theta}\right) = \frac{\partial^2 f}{\partial \theta^2} - \Gamma_{11}^1 \frac{\partial f}{\partial \theta}$$
and the formula for $\Gamma_{11}^1$
$$ \Gamma_{11}^1 = \frac{1}{2} g^{11}\partial_1 g_{11}= \frac{a^2\cos\theta -b^2\sin\theta}{a^2\sin^2\theta + b^2 \cos^2\theta}. $$