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?