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I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem:

Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, simply connected manifold with everywhere nonpositive sectional curvature) and $Y$ is a complete totally geodesic submanifold of $X$. Let $NY$ be the normal bundle of $Y$ in $X$, and let $\exp_{NY}: NY\rightarrow X$ be the restriction of the exponential map to $NY$. Then $\exp_{NY}$ is a diffeomorphism. This is Theorem 2.5 of Chapter X.

I'm confused about a detail in the proof of the preceding Theorem, 2.4. Fix $y_0\in Y$, and for each $y\in Y$, let $P_{y_0}^y$ denote parallel transport from $T_{y_0}X$ to $T_{y}X$ along the unique geodesic connecting $y_0$ to $y$. Now, define the smooth map $E:Y\times N_{y_0}Y\rightarrow X$ by $E(y,v)=\exp_y(P^y_{y_0}v)$.

At the bottom of page 273, Lang claims that $dE_{(y,v)}(z,0)$ is orthogonal to $dE_{(y,v)}(0,w)$ for any $y\in Y,v\in N_{y_0}Y, z\in T_yY,w\in N_{y_0}Y$.

I don't understand why this is true. He says it follows from the Gauss Lemma (Lemma VIII.5.6), but I can't figure out how it follows. I'm probably missing something simple. Does anyone know why those things are orthogonal?

Also, Lang is the only reference I can find for the fact that $\exp_{NY}:NY\rightarrow X$ is a diffeomorphism. Does anyone know if this exists elsewhere in the literature?

Thanks.

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    $\begingroup$ This is a basic fact about Jacobi fields. I recommend reading about Jacobi fields in books such as Cheeger and Ebin's Comparison Theorems in Riemannian Geometry (which probably has a proof of the Cartan-Hadamard Theorem too) or other more modern books on Riemannian geometry. $\endgroup$
    – Deane Yang
    Commented Dec 8, 2015 at 2:17
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    $\begingroup$ @Deane Yang So, I accidentally asked this question from a temporary account... Would you mind explaining a bit how this comes from the basic theory of Jacobi fields? I figured one could do this by looking at two Jacobi fields $J_1,J_2$ along $t \mapsto \exp_y(tP^y_{y_0}v)$, one given by varying $y$ in the direction $z$ and the other given by varying $v$ in the direction $w$. Then we want to show that $F(t)=\langle J_1(t), J_2(t) \rangle$ is identically 0. $F(0)=0$ and $F'(0)=0$, so suffices to show $F''(t)\equiv 0$. $F''(t)$ is an expression involving the curvature, and isn't obviously 0... $\endgroup$ Commented Dec 8, 2015 at 20:46
  • $\begingroup$ Let $\Gamma(\cdot, t)$ a family of geodesics, so that for each $t$, $\Gamma(\cdot,t)$ is a geodesic. Also, assume that $\Gamma(\cdot,t)$ is parameterized by arclength. Let $S = \partial_s\Gamma(s,t)$ (which has length1) and $T = \partial_t\Gamma(s,t)$. Now differentiate $S\cdot T$ a couple of times with respect to $s$. Now use that partials commute and the definition of Riemann curvature, as well as its symmetries. $\endgroup$
    – Deane Yang
    Commented Dec 8, 2015 at 22:01
  • $\begingroup$ @Deane Yang This will show that if $S(0,0)\cdot T(0,0)=0$ and $S(0,0)\cdot \frac{D}{\partial s}T(0,0)=0$, then $S(s,0)\cdot T(s,0)=0$ for all $s$, but this isn't what I need in this case (unless I'm missing something.) In my case, I have two families of geodesics $\Gamma_1(s, t) = E(y, s(v+tw)) $, and $\Gamma_2(s,t) = E(\beta(t),sv)$, where $\beta$ is a curve through $y$ with tangent vector $z$. I want to show that $ T_1(s,0)\cdot T_2(s,0) = 0$ for all $s$, where $T_i = \partial_t\Gamma_i(s,t)$. $\endgroup$ Commented Dec 8, 2015 at 23:25
  • $\begingroup$ Sorry, I didn't look at the result carefully enough. I also don't see why the two Jacobi fields need to remain orthogonal. It seems to me that you have to prove that a set of linearly independent Jacobi fields remain linearly independent along a geodesic. I'm pretty sure this is worked out very nicely here: Heintze, Ernst; Karcher, Hermann A general comparison theorem with applications to volume estimates for submanifolds. Annales scientifiques de l'École Normale Supérieure, Sér. 4, 11 no. 4 (1978), p. 451-470 $\endgroup$
    – Deane Yang
    Commented Dec 9, 2015 at 18:25

1 Answer 1

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I'm still not sure about that detail in the proof in Lang, but here's a slightly different proof that $\exp_{NY}:NY\rightarrow X$ is a diffeomorphism.

Step one: $\exp_{NY}$ is surjective. To see this, let $x\in X\setminus Y$. Let $B$ be a closed metric ball centered at $x$ containing some point of $Y$. Since $X$ is complete, $B$ is compact, and since $Y$ is closed in $X$, $B\cap Y$ is compact. Thus, there is $y_0\in B\cap Y$ which is closest to $x$, and so we have $d(x,y_0) = d(x, B\cap Y) = d(x, Y)$. By differentiating $y\mapsto d(x,y)$ along any curve in $Y$ through $y_0$, we find that the geodesic connecting $y_0$ to $x$ is normal to $Y$ at $y_0$. In other words, if $\gamma:[0,1]\rightarrow X$ is the geodesic with $\gamma(0)=y_0$, $\gamma(1)=x$, then $\gamma'(0)\in NY_{y_0}$. So $\exp_{NY}(\gamma'(0))=x$, and we have surjectivity.

Step two: $\exp_{NY}$ is injective. Suppose there were distinct $v_0, v_1 \in NY$ such that $\exp_{NY}(v_0) = \exp_{NY}(v_1)=x$. If $v_0,v_1$ were based at the same point of $Y$, then the exponential map based at that point would fail to be injective, contradicting the Cartan-Hadamard theorem. So $v_1, v_2$ are based at distinct points $y_1,y_2\in Y$. Let $\gamma:[0,1]\rightarrow Y$ be the geodesic with $\gamma(0)=y_0, \gamma(1)=y_1$. Then we have $\frac{d}{dt}|_{t=0}d(x,\gamma(t)) = \frac{d}{dt}|_{t=1}d(x, \gamma(t))=0$, because the geodesic connecting $y_0$ to $x$ is normal to $\gamma'(0)$ at $y_0$ and the geodesic connecting $y_1$ to $x$ is normal to $\gamma'(1)$ at $y_1$. This contradicts the convexity of the distance function of $X$.

So $\exp_NY$ is bijective. As Deane pointed out in the comments, it now suffices to show

Step three: The differential of $\exp_{NY}$ is everywhere injective. For this, fix $y_0\in Y$, and for each $y\in Y$, let $P_{y_0}^{y}$ denote parallel transport from $y_0$ to $y$. Then the map $Y\times NY_{y_0} \rightarrow NY$ given by $(y, v)\mapsto P_{y_0}^y v$ is a diffeomorphism, so it suffices to show that the map $E$ defined as the composition $Y\times NY_{y_0} \rightarrow NY \stackrel{\exp_{NY}}{\rightarrow}X$ has everywhere injective differential. To this end, choose $(y,v)\in Y\times NY_{y_0}$ and nonzero $(z,w)\in T_yY\times NY_{y_0}$. We need to show $dE_{(y,v)}(z,w)\neq 0$. Let $\gamma(t)$ be the geodesic passing through $y$ at time 0 with velocity $z$. We need to show that $\frac{d}{dt}|_{t=0} \exp_{\gamma(t)} P_{y_0}^{\exp(\gamma(t))} (v+tw)$ is not zero. Let $\alpha(s)= \exp_y P^{y}_{y_0}(sv)$. Let $\Gamma(s,t) = \exp_{\gamma(t)} P_{y_0}^{\exp(\gamma(t))} s(v+tw)$, and $J(s) = \frac{\partial}{\partial t} \Gamma(s,0)$. $J$ is a Jacobi field along the geodesic $\alpha$. Set $f(s) = \langle J(s), J(s) \rangle$. We will show $f(s)>0$ for $s>0$. Setting $s=1$ will then yield the desired result.

Compute \begin{align*} f'(0) &= 2\langle \frac{D}{\partial s}\frac{\partial}{\partial t}\Gamma(0,0), \frac{\partial}{\partial t}\Gamma(0,0) \rangle\\ &= 2\langle \frac{D}{\partial t}\frac{\partial}{\partial s}\Gamma(0,0), \frac{\partial}{\partial t}\Gamma(0,0) \rangle\\ &= 2\langle \frac{D}{\partial t}|_{t=0}P_{y_0}^{\exp(\gamma(t))} (v+tw), \frac{d}{dt}|_{t=0}\gamma(t) \rangle\\ &=2 \frac{\partial}{\partial t}|_{t=0} \langle P_{y_0}^{\exp(\gamma(t))} (v+tw), \frac{d}{dt}\gamma(t) \rangle\\ &=0, \end{align*} since $P_{y_0}^{\exp(\gamma(t))} (v+tw) \in NY$ and $\frac{d}{dt}\gamma(t)\in TY$ for all $t$. Also, \begin{align*} f''(s) &= 2\langle J'(s), J'(s) \rangle + 2\langle J''(s), J(s) \rangle\\ &= 2\langle J'(s), J'(s) \rangle - 2\langle R(\alpha'(s), J(s))\alpha'(s), J(s) \rangle\\&\geq 0. \end{align*}

Now, $J(0) = z$, so if $z\neq 0$, then $f(0)>0$, which combined with the two computations above, proves $f(s)>0$ for all $s>0$. If $z=0$, then $w\neq 0$ and $J'(0)=w$, so $f''(0)>0$, and again we conclude $f(s)>0$ for all $s>0$.

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  • $\begingroup$ I haven't looked at your proof carefully, but it looks like what I had in mind. I recommend, when possible, figuring out your own proofs like this. The paper by Heintze and Karcher, as well as other papers by Karcher, helped me a lot when learning this stuff. I recommend looking at the papers by Jost and/or Karcher cited here: ams.org/journals/tran/1992-333-01/S0002-9947-1992-1052910-2/… $\endgroup$
    – Deane Yang
    Commented Dec 10, 2015 at 14:07

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