Suppose $ F $ is a homogeneous polynomial function of degree $ m $ on Euclidean space $ \mathbb{R}^{n+1} $. Restrict $ F $ to the unit sphere $ S^n $ and we get a function on $ S^n $ denoted by $ f $, prove:
$(1)$ $ |\nabla_{S^n}f|^2=|\nabla_{\mathbb{R}^{n+1}}F|^2-m^2f^2 $;
$(2)$ $ \Delta_{S^n}f=\Delta_{\mathbb{R}^{n+1}}F-m(m-1)f-mnf .$
Where $ \nabla_{S^n}, \nabla_{\mathbb{R}^{n+1}} $ are gradients on $ S^n $ and $ \mathbb{R}^{n+1} $ respectively, $ \Delta_{S^n}, \Delta_{\mathbb{R}^{n+1}} $ are Laplace operators on $ S^n $ and $ \mathbb{R}^{n+1} $ respectively.
Hint: Use Euler's homogeneous function theorem $ \langle (\nabla_{\mathbb{R}^{n+1}}F)_z, z \rangle=mF(z) $.
The question above comes from my Riemannian geometry textbook. Can someone give me a hint about how to deal with $ \nabla_{S^n}f $ ? I can't find any direct relation between $ \nabla_{S^n}f $ and $ \nabla_{\mathbb{R}^{n+1}}F $. Though we can use local coordinates to expand $ \nabla_{S^n}f $ and compute directly, I am still looking forward to a more elegant way to do this.
Assume we have a Riemannian manifold $(M,g)$ and a submanifold $N \subseteq M$. Denote by $h$ the induced Riemannian metric on $N$ (the pullback of $g$ under the inclusion map $i \colon N \hookrightarrow M$). Given a smooth function $F \colon M \rightarrow \mathbb{R}$, we can consider $f = F|_{N}$ which is a smooth function on $N$. Given $p \in N$, what is the relation between $(\nabla_M F)(p)$ and $(\nabla_N f)(p)$?
Denote by $P \colon T_pM \rightarrow T_p N$ the orthogonal projection onto $N$. Then $(\nabla_N f)(p) = P((\nabla_M F)(p))$ because $$ df|_p(v) = d(F \circ i)|_p(v) = dF|_p(di|_p(v)) = \left< (\nabla_M F)(p), di|_p(v) \right>_g = \left< di|_p \left( P((\nabla_M F)(p)) \right), di|_p(v) \right>_g = \left< P((\nabla_M F)(p)), v \right>_h $$
for all $v \in T_pN$.
In your case, $M = \mathbb{R}^{n+1}$ with the standard Euclidean metric and $N = S^n$ has codimension one. In addition, given $p \in S^n$, we have the orthogonal direct sum decomposition
$$ T_p(M) = T_p(S^{n}) \oplus \operatorname{span}_{\mathbb{R}} \{ p \} $$
where we think of $p$ both as a point in $\mathbb{R}^{n+1}$ and a tangent vector at $T_p(\mathbb{R}^{n+1})$. Hence,
$$ P(v) = v - \left<v, p \right>p $$
and so
$$ (\nabla_{S^n} f)(p) = (\nabla_{\mathbb{R}^{n+1}} F)(p) - \left< (\nabla_{\mathbb{R}^{n+1}} F)(p), p \right> p, \\ \| (\nabla_{\mathbb{R}^{n+1}} F)(p) \|^2 = \| (\nabla_{S^n} f)(p) \|^2 + \left| \left< \nabla_{\mathbb{R}^{n+1}} F)(p), p \right> \right|^2 = \| (\nabla_{S^n} f)(p) \|^2 + m^2 |f(p)|^2.$$
This handles the first part. I'll leave the second part to you.