# Doubt about Lee's proof of Gauss Lemma (first edition)

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?

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 $$\partial B_{|V|}(0)$$.