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I've read a bunch of books about differential geometry and topology, but some questions came up about the nature of vector fields and their connection to smooth manifolds. Suppose that there is a certain vector field V defined at each point p in a smooth manifold M, that is homeomorphic to another manifold N. Few questions:

  • What is the transformation law from vector field V on M to the vector field W on N?
  • How will the lenght of a vector v in M change, when there is a new metric tensor in N [is there any rule of transition]?

Thank you.

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    If you assume the existence of a diffeomorphism $\phi : M \to N$, you can define $W$ using the differential of $\phi$, i.e. $W \colon = d\phi \circ V$. If you have a Riemannian metric $g_N$ on $N$, then the length of $W$ is $g_N(W,W) = g_N(d\phi(V), d\phi(V))$. Note that if there is a metric $g_M$ on $M$ and if $\phi$ is also an isometry, then you have: $g_N(W,W) = g_M(V,V)$. – 2017-01-30
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    Thank you, your insight into isometry is life saving. I've just found about f-relatedness between diffeomorfic manifolds and in fact an interesting characteristic is that this relation between vetor fields is unique, according to the 6.5 theorem on [link](https://en.wikibooks.org/wiki/Differentiable_Manifolds/Diffeomorphisms_and_related_vector_fields). – 2017-01-30

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