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!