I'm having trouble understanding a proof of the Bishop's volume comparison theorem and any help would be really appreciated. It's a simple part of the proof but I'm not quite getting what they want to say. The proof is the one in Gallot, Hulin and Lafontaine's Riemannian Geometry book. So it starts something like this:
We take a geodesic $c(t)=\text{exp}_m(tu)$ with inicial point $m \in M$. Also take an o.n.b., $\, \{u,e_2,\cdots , e_n\}$, of $\,T_mM$ (the tangent space of M at m), and parallel vector fields , $(E_i)_{2\leq i\leq n}$, along $c(t)$ such that $E_i(0)=e_i$. Now, given the correct restrictions so $T_{ru}\,\text{exp}_m$ (the differential of $\text{exp}_m$ at $ru\in T_mM$) is a isomorphism we can find Jacobi fields staisfying:
$$Y_i^r(0)=0,\; Y_i^r(r)=E_i(r)$$ The specific form of these Jacobi fields is: $$Y_i^r(t)=T_{tu}\text{exp}_m(tv) $$
Here we are making the identification $T_{ru}T_mM \approx T_mM$, and $v \in T_mM$ is the unique vector such that $T_{ru}\text{exp}_m(rv)=E_i(r)$. So nothing worng with the setting until now, but then the proof reads:
$``$Now,
$$J(u,t)=C_r t^{1-n}\text{det}(Y_2^r(t),\dots,Y_n^r(t)) $$ $$\text{where } C_r^{-1}=\text{det}((Y_2'^{r}(0),\dots,Y_n'^{r}(0)) )"$$ Here $J(u,t)= t^{1-n}\text{ det}(g(\tilde Y_i(t),\tilde Y_j(t)))^{1/2}$ with $\tilde Y_i(t)$ being the Jacobi Field that satisfies $\tilde Y_i(0)=0$ and $\tilde Y_i'(0)=e_i$ for $2\leq i \leq n$.
I totally don't get this statement, why are these equalities valid? I don't even understand what do they mean by $\text{det}(Y_2^r(t),\dots,Y_n^r(t)) $ since thay are already using another notation for the determinant of the metric. If anyone can help me clarify this it would be extremelly nice.
Lastly, I would like to know if anybody can recomend another nice reference for a proof of this theorem or the Gromov Volume comparison theorem. Thanks!
Before we do the volume comparison, we can first do the volume density comparison. To do that, we can choose an orthonormal basis $\{\frac{\partial}{\partial r}, v_1,\cdots, v_{n-1}\}$ in $T_xM,$ such that $v_i=\dot{Y}_i(0)$, where $\{Y_1,\cdots, Y_{n-1}\}$ are Jacobi fields along geodesic $\gamma(t)=\exp_x(t\frac{\partial}{\partial r})$. According to the Gauss Lemma, $$\{d\exp_{t\frac{\partial}{\partial r}}(\frac{\partial}{\partial r}), d\exp_{t\frac{\partial}{\partial r}}(v_1),\cdots d\exp_{t\frac{\partial}{\partial r}}(v_{n-1})\}$$ is an orthonormal basis in $T_{\gamma{(t)}}M$, and $d\exp_{t\frac{\partial}{\partial r}}(\frac{\partial}{\partial r})=\dot{\gamma}(t)$, and $d\exp_{t\frac{\partial}{\partial r}}(v_i)=t^{-1}Y_i(t)$. The Jacobian of exponential map is given by \begin{align*} \left|d\exp_{t\frac{\partial}{\partial r}}(\frac{\partial}{\partial r})\wedge \cdots d\exp_{t\frac{\partial}{\partial r}}(v_{n-1})\right|&=|\dot{\gamma}(t)\wedge t^{-1}Y_1(t)\wedge\cdots t^{-1}Y_{n-1}(t)|\\ &=t^{1-n}|Y_1(t)\wedge\cdots\wedge Y_{n-1}(t)|. \end{align*} I hope this gives you an idea how the identity works.
BTW, you can refer Schoen & Yau's book of Lectures on Differential geometry.