# How did the exponential map of Riemannian geometry get its name?

I've read in several books, including Milnor's Morse Theory and Petersen's Riemannian Geometry, that the exponential map in Riemannian geometry is named so because it agrees with the exponential map in Lie theory, at least for a certain choice of metric on the Lie group.

Is this the real reason why Riemannian geometers originally called the exponential map by that name? Or does the exponential map of Riemannian geometry fulfill some relation of the form $f=f'$ that characterizes "exponential" functions? The thing with Lie groups seems to be a highly nontrivial theorem and is not really covered in standard introductions to the subject like do Carmo and Lee.

• Do you know how to compute $\exp(A)$ where $A$ is a matrix? See if the "exponential map" of a Lie group of matrices has anything to do with this $\exp(A)$ computation. Mar 19, 2016 at 21:43
• @GeraldEdgar Uh, my whole post is asking if that exact operation is the reason why the map is called that. (For a matrix Lie group, the exponential map and matrix exponential are the same. That's presumably where the Lie group exponential gets its name from.) Mar 19, 2016 at 21:47

Nor does one find "exponential map" or "matrix exponential" (or matrix groups for that matter) in Lie's various papers and monographs of 1880-90s on continuous groups. He was talking about groups of transformations acting on domains of $\mathbb{R}^n$, and his "infinitesimal transformations" generating them were vector fields. He does write expressions like $f+X(f)+\frac1{2!}X^2(f)+\cdots$, but without symbology or much fuss about convergence. Schur added the last bit around 1890, when he gave a more rigorous proof of Lie's theorem on reconstructing Lie group from its Lie algebra, along with implicit version of the Campbell-Hausdorff formula for $Z$ in $e^Xe^Y=e^Z$. But only Campbell in two short notes of 1897 on that formula invokes exponentials directly:"If x and y are operators which obey the ordinary laws of algebra, we know that $e^ye^x = e^{y+x}$. I propose to investigate the corresponding theorem when the operators obey the distributive and associative laws, but not the commutative". Poincare arrived at the same ideas independently, and developed the exponential formalism more fully in On Continuous Groups (1899), where he also introduced what is now called "universal enveloping algebra" (rigorous proof of the formula by modern standards was only given by Hausdorff in 1906). See Schmid's lecture Poincaré and Lie groups, which cites the original sources.