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Let $M$ be a compact Riemannian manifold, $dvol_M$ be the volume form associated to the Hermitian metric $h^{TM}$, and $a$ the injectivity radius of $M$. We have the identification: $v \in T_xM, |v| \le \frac{a}{2} \mapsto exp_x(v) \in M$. Here are my questions:

1) Why does there exist a smooth function $k'_x(v)$ such that: $dvol_M(v) = k_x'(v)dvol_{TM}(v)$

2) Why can we assume, for $\epsilon_0$ small enough so that $|v| \le 4\epsilon_0, v \in T_xM$ that:
$\frac{1}{2} h^{TM}_x \le h^{TM}_v \le \frac{3}{2} h^{TM}_x$

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These questions follow from the basic properties of the exponential map, which is the same thing as geodesic normal co-ordinates. You should be able to find this in any introductory book to Riemannian geometry, such as the ones by Jost or Gallot-Hulin-Lafontaine or Do Carmo. – Deane Yang Aug 10 '12 at 18:40
Thank you. I will take a look in these books. I am just a beginner in this topic. – mena Aug 11 '12 at 13:34

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