Periodic and Nonconstant Integral Curve: A Contradiction

[huyichen]

M is a smooth manifolds, and X is a vector field on M, y is a maximal integral curve of X. Now suppose y is periodic and nonconstant, show that there exists a unique positive number T(called the period of y) such that y(t)=y(t') if and only if t-t'=kT for some integer k.(For this problem, What is the contradiction you will get when you assume the positive number is not unique?)
Also show that the image of y is an immersed submanifold of M, diffeomorphic to R, S^1, or R^0. (Have no idea)
Actually this is 17-5 from Introduction to smooth manifold Lee's book, just in case you are not clear about the any concepts in this problem.

 

[pat_connell]

For the first part, assume that there is not a unique positive number T. Then there exists two different positive numbers T_1 and T_2 such that y(t)=y(t') if and only if t-t'=kT_1 or kT_2 for some integer k. Combining these two equations, we get t-t'=(kT_1)/l = (kT_2)/l for some integers l and k. This is a contradiction, since the left side of the equation is an integer while the right side is not. Therefore, there must exists a unique positive number T.For the second part, since y is periodic and nonconstant, it is a surjective continuous map from R to M. Since y is also an immersion, its image is an immersed submanifold diffeomorphic to R, S^1, or R^0.