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In the proof of Gauss's lemma here, there is a step

$\displaystyle\lim_{t\to0}\frac{\partial f}{\partial s}(0,t)=\lim_{t\to0}T_{tv}\ \exp_p(tw_N)=0$

However, the limit seems meaningless (unless tangent bundle has been introduced) since $T_{tv}\ \exp_p(tw_N)$ lies in different tangent spaces for different $t$. Instead I think one can directly conclude that

$\displaystyle\frac{\partial f}{\partial s}(0,0)=T_{0}\ \exp_p(0)=0$

since $f(0,s)$ is a constant curve.

Am I right?

(This step is also present in do Carmo's Riemannian Geometry.)

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You do not need to introduce the tangent bundle to make sense of the limit. It is enough to choose a coordinate patch, because that induces a canonical identification of all the tangent spaces in the patch. –  Zhen Lin Mar 26 '12 at 12:03
    
Yes, so my question should focus on: is the argument using limit unnecessarily complicated? –  Junyan Xu Mar 27 '12 at 7:18

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