I am reading the John Lee's Introduction to Riemannian manifolds, p.332, proof of the theorem 11.11 and stuck at some statement.
Theorem 11.11 ( Laplacian comparison I ) Suppose $(M,g)$ is a Riemannian $n$-manifold whose sectional curvatures are bounded above by a constant $c$. Suppose $p \in M$, $U$ is a normal neighborhood of $p$, $r$ is the radial distance function on $U$, and $s_c$ is defined as in Proposition 11.3. Then on $U_0-\{p\}$, we have
$$\Delta r \ge (n-1) \frac{s_c'(r)}{s_c(r)},$$
where $U_0 =U $ if $c \le 0$, while $U_0 =\{q\in U : r(q) < \pi R \}$ if $c= 1/R^2 >0.$
Proof. By the result of Problem 5-14, $\Delta r= \operatorname{tr}_g(\nabla^2 r) = \operatorname{tr}(\mathcal{H}_r).$ The result then follows from the Hessian comparison theorem, using the fact that $\operatorname{tr}(\pi_r) =n-1$, which can be verified easily expressing $\pi_r$ locally in an adapted orthonormal frame for the $r$-level sets. QED.
Here, for each $q\in U-\{p\}$, $\pi_r : T_qM \to T_qM$ is the orthogonal projection onto the tangent space of the level set of $r$ ( equivalently, onto the orthogonal complement of $\partial_r|_q$ ; the radial vector field ). ( C.f. Statement of Proposition 11.3. in his book ) ; i.e., $\pi_r (w) = w- \langle w, \partial_r \rangle \partial_r $. If needed, I will upload more information, for example , radial distance function, $s_c$, Hessian comparion theorem, etc..
I don't understand the bold statement at all. First, I don't know meaning of each terminologies. What is defitnition of $r$-level sets? What the sentence 'expresstion of $\pi_r$ locally in an adapted orthonormal frame for the $r$-level sets' exactly means? What mechanism would works to prove the $\operatorname{tr}(\pi_r) =n-1$?
Can anyone help?
Best Answer
This is a linear algebra fact. If $p\colon E \to E$ is a projector, where $E$ is finite dimensional, then its trace is equal to its rank. Indeed, let $F\subset E$ be its image, and let $G$ be its kernel. Then $E = F \oplus G$, and in an adapted basis, the matrix of $p$ is $$ \begin{pmatrix} Id_F & 0 \\ 0 & 0 & \end{pmatrix}, $$ so that the trace of $p$ is the dimension of $F$, that is, its rank. Here, the projector $p$ is a projector onto an hyperplane in dimension $n$, so its trace is $n-1$.
As for the terminology: the $r$-level set of a function $f$ is the set $f^{-1}(r) = \{x \mid f(x) = r\}$. I guess here the function is the distance function from some point. The $r$-level set above $p$ is thus the geodesic sphere at distance $r$.