I am reading the John Lee's Introduction to Riemannian manifold ( 2nd Edition ), Theorem 11.7 ( Hessian Comparison ) and stuck at some statement.
First I arrange some relavant definitions and theorems.
Definition 1. Given a finite dimensional inner product space $V$ and self-adjoint endomorphism $A,B : V \to V$, the notation $A \le B$ means that $\langle Av ,v \rangle \le \langle Bv ,v \rangle$ for all $v\in V$, or equivalently that $B- A$ is positive defitinte.
Theorem 11.5 ( Riccati Comparison Theorem ). Suppose $(M,g)$ is a Riemannian manifold and $\gamma : [a,b] \to M$ is a unit-speed geodesic segment. Suppose $\eta$, $\tilde{\eta}$ are self-adjoint endomorphism fields along $\gamma_{(a,b]}$ that satisfy the following Riccati equations :
$$ D_t \eta + \eta^2+\sigma=0, D_t\tilde{\eta}+\tilde{\eta}^2+\tilde{\sigma}=0,$$
where $\sigma$ and $\tilde{\sigma}$ are continuous self-adjoint endomorphism fields along $\gamma$ satisfying $$\tilde{\sigma}(t) \ge \sigma(t) \ \operatorname{for all} t\in [a,b] $$
Suppose further that $\lim_{t \searrow a} ( \tilde{\eta}(t) - \eta(t))$ exists and satisfies
$$ \lim_{t \searrow a}(\tilde{\eta}(t)-\eta(t)) \le 0.$$
Then $$\tilde{\eta}(t) \le \eta(t) \ \operatorname{for all} t\in (a,b]. $$
Definiton 2. For each $c\in \mathbb{R}$, let us define a function $s_c : \mathbb{R} \to \mathbb{R}$ by
$$ s_c(t) = \begin{cases} t, \ \operatorname{if} c=0 ; \\ R \operatorname{sin} \frac{t}{R}, \ \operatorname{if} c= \frac{1}{R^2}>0 ; \\ R \operatorname{sinh} \frac{t}{R} , \ \operatorname{if} c= - \frac{1}{R^2} <0 . \end{cases} $$
Definition 3. Suppose $(M,g)$ is a Riemannian manifold, $U\subseteq M$ is a normal neighborhood of $p\in M$, and $r$ is the radical distance function on $U$. 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 $\partial_r|_q$).
Proposition 11.3. Suppose $(M,g)$ is a Riemannian manifold, $U\subseteq M$ a normal neighborhood of $p\in M$, and $r$ is the radial distance function on $U$. Then $g$ has constant sectional curvature $c$ on $U$ if and only if the following formula holds at all points of $U - \{ p\}$ : $$ \mathcal{H}_r = \frac{s_c'(r)}{s_c(r)}\pi_r$$,
Theorem 11.4 ( The Riccati Equation ). Let $(M,g)$ be a Riemannian manifold; let $U$ be a normal neighborhood of a point $p\in M$ ; let $r : U \to \mathbb{R}$ be the radial distance function ; and let $\gamma : [0,b] \to U$ be a unit-speed radial geodesic. The Hessian operator $\mathcal{H}_r$ satisfies the following equation along $\gamma|_{(0,b]}$, called a Riccati equation : $$ D_t \mathcal{H}_r + \mathcal{H}_r^{2} + R_{\gamma'}=0,$$
where $\mathcal{H}^2_r$ and $R_{\gamma'}$ are the endomorphism fields along $\gamma$ defined by $\mathcal{H}_r^2(w)=\mathcal{H}_r ( \mathcal{H}_r(w))$ and $R_{\gamma'}(w)=R(w,\gamma')\gamma'$, with $R$ is the curvature endomorphism of $g$.
Now I state Hessian comparison theorem.
Theorem 11.7 ( Hessian Comparison ) Suppose $(M,g)$ is a Riemannian $n$-manifold. $p\in M$. $U$ is a normal neighborhood of $p$, and $r$ is the radial distance function on $U$.
(a) If all sectional curvature of $M$ are bounded above by a constant $c$, then the following inequality holds in $U_0 - \{p\}$ :
$$ \mathcal{H}_r \ge \frac{s_c'(r)}{s_c(r)}\pi_r,$$
where $s_c$ and $\pi_r$ are defined as in Proposition 11.3 (above definitions), and $U_0=U$ if $c\le 0$, while $U_0 = \{q\in U : r(q) <\pi R \}$ if $c= 1/R^2 >0$.
(b) If all sectional curvature of $M$ are bounded below by a constant $c$, then the following inequality holds in all of $U-\{p\}$ : $$ \mathcal{H}_r \le \frac{s'_c(r)}{s_c(r)}\pi_r.$$
Proof of the Theorem 11.7. Let $(x^1,\dots, x^n)$ be Riemannian normal coordinates on $U$ centered at $p$, let $r$ be the radial distance function on $U$, and let $s_c$ be the function defined by $(10.8)$ ( c.f. above definition ). Let $U_0 \subseteq U$ be the subset on which $s_c(r) >0$;when $c\le 0$, this is all of $U$, but when $c= 1/R^2 >0$, it is the subset where $r<\pi R$. Let $\mathcal{H}_r^c$ be the endomorphism field on $U_0 - \{p\}$ given by $$ \mathcal{H}_r^{c}=\frac{s_c'(r)}{s_c(r)}\pi_r$$
Let $q\in U_0-\{p\}$ be arbitrary, and let $\gamma : [0,b] \to U_0$ be the unit-speed radial geodesic from $p$ to $q$. Note that at every point $\gamma(t)$ for $0 < t \le b$, the endomorphism field $\pi_r$ can be expressed as $\pi_r(w) =w-\langle w, \partial_r \rangle \partial_r = w - \langle w,\gamma' \rangle \gamma'$, and in this form it extends smoothly to an endomorphism field along all of $\gamma$. Moreover, since $D_t\gamma' =0$ along $\gamma$, it follows that $D_t\pi_r =0$ along $\gamma$ as well. Therefore, direct computation using the facts that $s_c''=-cs_c$ and $\pi_r^2=\pi_r$ shows that $\mathcal{H}_r^c$ satisfies the following Riccati equation along $\gamma|_{(0,b]}$ :
$$ D_t \mathcal{H}_r^c + (\mathcal{H}_r^c)^2 +c\pi_r =0.$$
( Next proof is omitted ).
I am trying to understand the bold statement. Why $\partial_r = \gamma'$ ? And what the sentence "and in this form it extends smoothly to an endomorphism field along all of $\gamma$" exactly means? And finally why $D_t\pi_r=0$ along $\gamma$?
EDIT ( My first trial ) : I think that $\partial_r = \gamma'$ is verified in Chris's answer in Existence of unit speed geodesics. Note that $D_t : \mathfrak{X}(\gamma) \to \mathfrak{X}(\gamma)$ is the covariant derivatie along $\gamma$. And I think that the statement "in this form it ( $\pi_r$ ) extends smoothly to an endomorphism field along all of $\gamma$ " means that $\pi_r$ extends smoothly to $\mathfrak{X}(\gamma) \to \mathfrak{X}(\gamma)$. True?
And I think that the statement $D_t \pi_r= 0$ means that the composition $D_t \circ \pi_r$ is zero. True?
And let me show that $D_t \circ \pi_r = 0.$ Fix $w \in \mathfrak{X}(\gamma)$. Then since $D_t \gamma' =0$,
$$ \begin{aligned} D_t \circ \pi_r (w) = D_t(w- \langle w, \gamma' \rangle \gamma' ) = D_t w - D_t(\langle w, \gamma' \rangle \gamma') = \\ D_tw - \{\frac{d}{dt}\langle w, \gamma' \rangle \gamma' + \langle w, \gamma' \rangle D_t \gamma' \}= D_t w - \frac{d}{dt}\langle w, \gamma' \rangle \gamma' = \\ D_t w - \{\langle D_t w , \gamma' \rangle \gamma' + \langle w, D_t \gamma' \rangle \gamma' \} = D_t w - \langle D_t w , \gamma' \rangle \gamma' \stackrel{?}{=} 0 \end{aligned} $$
For the fifth equality we used the John Lee's book Proposition 5.5 -(d) ; $(M,g)$ is endowed with the Riemannian connection.
Is this argument correct? Am I following the author's intension well? If so, the final equality is true? Why? I'm a little confused.
EDIT : After some considertaion, I think that perhaps, $D_t \pi_r$ means that "for each $w\in \mathfrak{X}(\gamma)$, $D_t \pi_r(w) := D_t (\pi_r(w)) - \pi_r(D_t (w))$."
Then since $ \pi_r ( D_t (w)) = D_t w - \langle D_t w , \gamma' \rangle \gamma'$, we are done. And I am not certainly sure that $D_t \pi_r(w) := D_t (\pi_r(w)) - \pi_r(D_t (w))$ until now.
Can anyone help?