Existence and Uniqueness of Geodesics I'm reading the theorem 4.10 of the book "Riemannian manifolds" by John Lee. In proof, page 58, the author refers to the existence and uniqueness theorem for first-order ODEs [Boo86, Theorem IV.4.1]. Also in page 60, theorem 4.12 is about the Existence and Uniqueness for Linear ODEs. I want to know if this two theorem are the same? Is there any difference between them? Thanks!
 A: The existence and uniqueness theorem for first-order ODE's is an important theorem that is discussed in many sources (although I don't have the book Lee refers to in front of me). A loose statement of the theorem is: 
A reasonable system of ODE's with specified initial conditions of the form $$y'(t) = f(t, y(t)), ~ y(0) = y_0$$
has a unique solution for at least a short time, i.e., for $t$ in some neighborhood of $0$.
("Reasonable" means that $f$ can be Lipschitz, for example.)
The standard proof uses Picard iteration and the contraction mapping theorem (a.k.a. the Banach fixed point theorem). See the Wikipedia article: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
Some comments:


*

*This theorem applies to nonlinear ODE systems, i.e., $f$ is allowed to depend nonlinearly on $y(t)$. (Note, though, that the equation is linear in $y'(t)$.)

*This theorem is about first-order ODE systems, but an ODE system of any order can be rephrased as a first-order system by introducing new variables if necessary. (It needs to be linear in the highest-order derivatives for the theorem to apply, though.)


Regarding the geodesic equation: the geodesic equation is a second-order nonlinear system to which this theorem applies, so given an initial condition, the geodesic equation has a unique solution for at least short time (this is Lee's Theorem 4.10). In the proof of Theorem 4.10, Lee makes explicit the rephrasing of the second-order system as a first-order system by introducing new variables.
Regarding Lee's Theorem 4.12: the assumptions are stronger than in the above general existence and uniqueness theorem because the system is now linear. The linearity means that a stronger result holds: rather than just short-time existence, solutions exist for all $t$. Lee uses this to prove Theorem 4.11 about parallel translation along curves. The parallel translation equation is linear, but the geodesic equation is not.
