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Question

So I'm from an engineering and recently learned about integrating on riemannian manifolds. However, I have been faced with the notation $$\int _{T_xM} f(z) \ dz,$$ where $M$ is a Riemannian manifold and $T_x$ is the tangent space at some point $x$. Note that I have learned that $T_xM$ is the space of all tangent vectors at $x$. Can anyone say what this integration notation means? I cannot find any references online.

I have been taught that if $M$ is a Riemannian manifold then $$\int_M f=\int_{\mathbb{R}^n}f \sqrt{\mathrm{det}(A^tA)} \ dx_1\ldots,dx_n$$ where $A$ is the jacobian (I think).

query
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    What are the function $f$ and the variable $z$? – Amitai Yuval Apr 17 '18 at 19:06
  • Well this is in the context of manifold learning so I think $f$ is really $f(x,z)$ (a gaussian kernel usually, so it is $C^{\infty}$ in both $x$ and $z$). As far as I know $z$ is a variable of integration...I have presented this as I saw it in class. There is a note that $z\in T_xM\subset \mathbb{R}^n$ has coordinates $(z,0)^T$ – query Apr 17 '18 at 19:21
  • Every vector space equipped with an inner product is a Riemannian manifold in its own right. Meaning that if you can understand $\int_Mf$, you can also understand $\int_{T_xM}f$. – Ivo Terek Apr 17 '18 at 19:47
  • So you are saying $T_xM$ has an inherited inner product? Ah I must have missed that... – query Apr 17 '18 at 20:01

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