Let $M \subset \mathbb{R}^n$ a subvariety of dimension $k$. How can I define the Jacobian of a smooth map $\phi :\mathbb{R}^t \rightarrow M$ where $t \leq k$?. It is useful to know that there exist a inner product on the tangent space on every point of $M$ ?
Jacobian of a smooth map
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differential-geometry
riemannian-geometry
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0You have to define the Jacobian up to a chart at a point. – 2017-01-06
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0And how can I do that? – 2017-01-06
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0It is like defining the Jacobian of $f:R^n\rightarrow R^m$. – 2017-01-06