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This question follows from another question Relations between two definitions of Lie algebra.

Exponential maps basically links (though not necessarily one one) a tangent vector to a point at a manifold interpreted as a 'distance'/displacement. With exponential maps we link tangent space (a vector space) to the manifold (made a Lie group).

But why we need to, with exponential maps, make a link between a tangent space and the manifold? i.e. what motivates the concept of exponential maps initially? I guess in physical context it enables us to calculate projection distance from initial projection speed, but in other context I don't know why.

Charlie Chang
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    The motivating concept of tangent space (e.g. "tangent line" and "tangent plane" in basic calculus) is that smooth manifolds (curves, surfaces, etc.) can be approximated by linear "flat" spaces. The exponential map from (a subset of the tangent space near zero) to a neighborhood of the manifold is just a manifestation of this idea of "linear approximation". – Nick Jul 26 '20 at 16:32
  • The example of $SO(2)$ ~ $S^1$, where $\exp(it)=\cos(t)+I\sin(t)$ which projects the point $it$ on tangent line exactly linearly to point $e^{it}$ on $S^1$ seems to illustrate the point. I guess there are examples of manifolds where the linearity holds to different extent. – Charlie Chang Jul 26 '20 at 16:55
  • Seems a reason is given by my own post: https://math.stackexchange.com/q/3771008/577710, that with exponential map $\exp(p)$ we get a 'closed curve' which is perpendicular to all the geodesics passing through $p$. So we can have 'closed curve' that very much resembles a circle, and a geodesic that resembles a radius or a straight line. This is like giving a projective geometry structure on a manifold. It also gives an example of tubular neighborhood, which is a neighborhood containing lines (e.g. geodesics) all perpendicular to a curve (e.g. the 'closed curve' given by $exp_p$) – Charlie Chang Jul 27 '20 at 18:24
  • About tubular neighborhood https://en.wikipedia.org/wiki/Tubular_neighborhood. PS: Though I am still not sure if that's the original motivation for this concept. – Charlie Chang Jul 27 '20 at 18:26

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