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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 closed totally geodesic submanifold of $X$. Let $NY$ be the normal bundle of $Y$ in $X$. Then the exponential map $\text{exp}_{NY}NY\rightarrow X$ 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 translation 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 $E(y,v) = \exp_y(P_{y_0}^yv)$.

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? Thanks.

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  • $\begingroup$ Did you solve your question. If so, please post it. Or if you have a file of book, can you link ? I have no book. But I want to read the general version of CH theorem (Since webpage does not contain this version) $\endgroup$ – HK Lee Apr 3 '16 at 8:42

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