Let $M$ be a smooth $n$-dimensional Riemannian mnaifold and $exp_p:B_r(0)\rightarrow B_s(p)$ be the differmorphism, for $p\in M$ and $0\in T_pM$. Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth function. Then I am confuse about the following integration formula
$$\int_{B_r(0)}f(v)dv=\int_{B_s(p)}f(exp^{-1}_px)|D_xexp^{-1}_p|dx.$$
I think I am doing some wrong in change of variable and I can not correct it. Please Help me by giving me the correct expression.
Thank you.
2026-03-25 06:03:29.1774418609
Change of variable of exponential map in a manifold
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