Umbilic hypersurfaces of the hyperbolic space

Question

Let $(M^{n+1},g)$ be a manifold with a Riemannian metric $g$ and let $\nabla$ be its Riemannian connection. We say an immersion $x : N^n \to M^{n+1}$ is (totally) umbilic if for all $p \in N$, the second fundamental form $B$ of $x$ at $p$ satisfies $$\langle B(X,Y),\eta \rangle(p)=\lambda(p)\langle X,Y \rangle,$$ for all $X,Y \in \mathcal X(N)$ and $\lambda(p) \in \mathbb R$ and for a given unit field $\eta$ normal to $x(N)$; here we are using $\langle \, , \, \rangle$ to dontate the metric $g$ on $M$ and the metric induced by $x$ on $N$.

e) Calculate the mean curvature and the sectional curvature of the umbilic hypersurfaces of the hyperbolic space.

(For the other parts (a)-(d), please see this link: Problem 8.6 of do Carmo, Riemannian Geometry. Nonetheless, my question is only on part (e), specifically on calculating the mean curvature.)

The author gave the following hint:

Hint: Consider the model of $H^n$ as the upper half-space. Let $S \cap H^n$ be the intersection of $H^n$ with a Euclidean $(n-1)$-sphere $S \subset \mathbb R^n$ of radius $1$ and center in $H^n$. Since $S \cap H^n$ is umbilic, all of the directions are principal, and it is enough to calculate the curvature of the curves of intersection of $S \cap H^n$ with the $x_1x_n$-plane. Use the expression obtained in part (b) of this exercise to establish that the mean curvature of $S \cap H^n$ (in the metric of $H^n$) is equal to $1$ if $S$ is tangent to $\partial H^n$, is equal to $\cos \alpha$ if $S$ makes an angle $\alpha$ with $\partial H^n$, and is equal to the "height" of the Euclidean center of $S$ relative to $\partial H^n$, if $S \subset H^n$. To calculate the secitonal curvature, use the Gauss formula.

The expression from part (b) I believe the author is referring to is $$ \left\langle \overline \nabla_X \left(\frac{\eta}{\sqrt{\mu}} \right),Y \right\rangle_{\overline g} = \frac{-2\lambda \mu+\eta(\mu)}{2\mu\sqrt{\mu}}\langle X,Y \rangle_{\overline g}. $$ I am not sure on how to apply this formula to establishing the mean curvature, depending on how $S$ is related to $\partial H^n$ ($S$ is tangent to $\partial H^n$, $S$ makes an angle $\alpha$ with $\partial H^n$, etc.).

Do Carmo defined earlier (page 129 to be exact) in the textbook the mean curvature to be $\frac 1n(\lambda_1+\cdots+\lambda_n)$, where $\lambda_i=k_i$ are principal curvatures of $f$.

I am sorry for being unable to show much of what I tried on this question. Nonetheless, any hint or first step will be helpful. Thanks!

Answer 1

I think it goes like this: You can write $B = \lambda g$ in coordinates, use Codazzi equation and the fact that covariant derivative of the metric vanishes ($\nabla g = 0$): For $i,j\neq k$: $$\nabla_{k} B_{ii} = \nabla_{i} B_{ki},$$ $$(\nabla_{k} \lambda) g_{ii} =(\nabla_{i} \lambda) g_{ki},$$ $$(\nabla_{k} \lambda) g_{ii}g^{ij} =(\nabla_{i} \lambda) g_{ki}g^{ij},$$

$$(\nabla_{k} \lambda) \delta_{i}^{j} =(\nabla_{i} \lambda) \delta_{k}^{j},$$ $$(\nabla_{k} \lambda)=0,$$

Then $\lambda$ is constant, the principal curvatures are equal, then mean curvature is either $0$ or $\lambda\neq0$.

On orthonormal coordinates we have Gauss equation as $$-1=\hat{R}_{ijji} = R_{ijji} + B_{ij}B_{ij} - B_{ii}B_{jj}=R_{ijji} -\lambda^2$$ from where the sectional curvature follows.

In order to follow the hint you have to use the metric on the upper half plane $$ds^2 = \frac{\sum_i (dx^i)^2}{x_{n}^2},$$ then use this hyperbolic distance as in the suggestion: $$\text{d}_h (x(p) , x_0)^2 = \frac{1}{\lambda^2},$$ where the distance is something like $$ \text{d}_h (x,y) = 2\,\text{arcsinh} \left(\frac{|x-y|}{2\sqrt{x_ny_n}}\right).$$

Add: Just to be clearer in the last part, consider the unit circle with centre on the vertical axis $(0,y)$, $(y>0)$, in the upper half plane with the metric given above. Then we can write $$x(p) = (\cos\theta, y+ \sin\theta),$$ $$\partial_x = (-\sin\theta, \cos\theta),$$ $$D_{\partial_x}\partial_x = (-\cos\theta, -\sin\theta),$$ $$\hat{n} = \sqrt{y+\sin\theta}(-\cos\theta, -\sin\theta)$$

and before doing the hyperbolic product $\langle D_{\partial_x}\partial_x,\hat{n} \rangle$ look at values of $y$ for which $\hat{n}$ degenerates.