A vector field is supposed to be an infinitesimal generator of a lie subgroup. Typically to generate a flow on a manifold using a one parameter group of transformations, we do exponentiation of the vector field. Does it have to be necessarily exponential or any other function which can generate the flow will do?
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2What other function do you have in mind? It is not like we have a choice here... – Mariano Suárez-Álvarez May 10 '14 at 08:02
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The vector field $X(p)=1$ has the flow $\phi_t(p)=p+t$, nothing exponential about it. – Lutz Lehmann May 10 '14 at 11:36