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I have a doubt about the proof of Gauss Lemma which appears in the first edition of Lee's book "Riemannian Manifolds: An Introduction to Curvature" (see Theorem 6.8, p.102-103, here).

The proof goes essentially as follows. We pick some arbitrary $q=\exp_p(V)$ in a geodesic ball $U$ centred at $p$ and a vector $X\in T_qM$, which is assumed to be tangent to the geodesic sphere through $q$. Then, since $\exp_p$ in a diffeomorphism onto the geodesic ball $U$, by identifying $T_VT_pM$ with $T_pM$, there exists a vector $W\in T_pM$ (which we imagine as emanating from $V$) such that $(D\exp_p)_V(W)=X$. We then claim that we can pick a curve $\sigma:(-\varepsilon,\varepsilon)\rightarrow T_pM$ such that $\sigma(0)=V$, $\sigma'(0)=W$ and $|\sigma(s)|=\text{const}=|V|$ and construct the variation $\Gamma(s,t)=\exp_p(t\sigma(s))$. We then proceed as usual by showing that $g(\partial_s\Gamma,\partial_t\Gamma)$ is independent of $t$, by proving $\frac{\partial}{\partial t}g(\partial_s\Gamma,\partial_t\Gamma)=0$. Note that for $\frac{\partial}{\partial t}g(\partial_s\Gamma,\partial_t\Gamma)=0$ to be true it is crucial that $|\sigma(s)|=\text{const}$.

My problem is: how can we guarantee that we can construct such a curve $\sigma$, which starts at $V$ with initial velocity $W$ and stays on the sphere $\partial B_{|V|}(0)\subset T_pM$, unless we already know that $W$ is perpendicular to $V$? And, unless I am missing something crucial here, we don't know that, because $W$ is merely the preimage of $X$ under the linear map $(D\exp_p)_V$. In some sense the fact that $W\perp V$ is the whole content of the lemma.

What am I missing?

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Fact: Given any manifold $N$, any $x\in N$, and any $w\in T_xN$, there exists a curve $\sigma:(-\varepsilon,\varepsilon)\to N$ such that $\sigma(0)=p$ and $\sigma'(0)=W$.

Using the fact with $N=\partial B_{|V|}(0),\ x=q$ and $w=W$ then you have the curve. Note that you only need $W$ to be a tangent vector of $T_p\partial B_{|V|}(0)$ for this curve to exist. This is assured by the following facts:

  1. $\exp_p$ is an diffeomorphism from $U'\to U$, where $U'$ is some subset of $T_pM\cong\mathbb R^n$.

  2. $X$ is tangent to the geodesic sphere $S_p\subset M$ through $p$.

  3. $W=(D\exp_p^{-1})X$ and $\partial B_{|V|}(0)=\exp_p^{-1}(S_p)$.

Hence $W$ is tangent to $\partial B_{|V|}(0)$.

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