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I am currently reading Getzler's book and I don't understand his definition of the Jacobian on page 38:

Let $M$ be a Riemannian manifold, $x_0\in M$ and consider the exponential map$$\exp_{x_0}\colon U\subset T_{x_0}M\to V\subset M$$ Suppose $\mathbf{x}\in U$ and $x:=\exp_{x_0}\mathbf{x}$ and consider the differential at $\mathbf x$:$$\mathrm{d}(\exp_{x_0})_{\mathbf{x}}\colon T_{x_0}M\to T_x M$$Now the Jacobian at $\mathbf{x}$ is defined as its determinant:$$j(\mathbf{x}):=|\det\mathrm{d}(\exp_{x_0})_{\mathbf{x}}|$$But the determinant is only defined for endomorphisms, isn't it?

Filippo
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    The determinant of a linear map $E \to F$ between two $n$-dimensional euclidean spaces is well defined as the determinant of its matrix in any pair of orthonormal bases: it is straightforward to show the independence from the chosen bases since they are orthonormal. Here, $\mathrm{d}(\exp_{x_0}){\mathbf{x}}$ is a linear map from $(T{x_0}M,g_{x_0})$ to $(T_xM,g_x)$ which are indeed euclidean. – Didier Mar 16 '22 at 12:58
  • The comment above answers my question. Note to myself: If $v_1,\ldots,v_n$ is an orthonormal basis of $T_{x_0}M$, then the local frame $\partial_1,\ldots,\partial_n$ defined by the associated normal coordinates satisfies $$\partial_{i,\mathbf{x}}=\mathrm{d}(\exp_{x_0})_{\mathbf{x}}v_i$$ and hence the definition is consistent with the equation $$j(\mathbf x)=|\det(e^j\partial_i)|$$ on page $38$, where $e_1,\ldots,e_n$ is some orthonormal basis of $T_xM$. – Filippo Mar 16 '22 at 21:07
  • @Didier Given two orthonormal bases of the same euclidean space, we only know that the absolute value of the determinant of the coordinate change is equal to $1$ (if we don't assume that they have the same orientation), so only the absolute value of the determinant is well defined, right? – Filippo Mar 16 '22 at 21:13
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    Sorry, I can't edit the above comment. But you have to consider positively oriented orthonormal bases of course in order to have the determinant. Otherwise, only its absolute value has meaning. Thank you for correcting me. – Didier Mar 16 '22 at 21:15
  • @Didier Thank you very much for the confirmation :) – Filippo Mar 16 '22 at 21:16

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