I've been studying differential geometry using Do Carmo's book. There's the notion of exponential map, but I don't understand why it is called "exponential" map. How does it has something to do with our common notion of exponentiation?
I read from the book The road to reality (by R. Penrose) that it is related to taking exponentiation when making (finite) Lie group elements from Lie algebra elements. It seems like using Taylor's theorem on a manifold so we have, for example, there was the following equation explaining why it is the case.
$f(t) = f(0) + f'(0)t + \frac{1}{2!}f''(0) t^2+\cdots = (1+t\frac{d}{dx}+\frac{1}{2!}t^2\frac{d^2}{dx^2}+\cdots)f(x)|_{x=0} = e^{t\frac{d}{dx}}f(x)|_{x=0}$.
The differential operator can be thought of as a vector field on a manifold, and it is how Lie algebra elements (which are vectors, on a group manifold (Lie group), in a tangent space at the identity element). If I understood correctly, the truth is that this is exactly the exponential map that sends a vector on a tangent space into the manifold in such a way that it becomes the end point of a geodesic (determined by the vector) having the same length.
Why is the above Taylor expansion valid on a manifold? Why is the exponential map the same as taking exponentiation?
