The words contained in this file might help you see if this file matches what you are looking for:

...Introduction to riemannian manifolds all will be connected hausdor and second countable terminology let m a smooth manifold denote the tangent space at x by txm if f n is map between associated on df tf i an open interval in r path then for t denotes d denition metric choice each point of positive denite inner product h products varying smoothly with known as we not give formal phrase local isometry two dieomorphism such that points vectors v w hv wi dh also manifoldand allows one dene vector length vi angle non zero cos lengths determine so which preserves necessarily rie mannian paths inherit given z dt this independent its parametrisation other words thisisjust consequence fact can change variable integration piecewise but nitely many construct y are inf from proposition does topology induced coincides original notation b crucial study notion geodesic here s standard equivalent usual speed where locally minimising means there e no shorter these figure exercise geodesics rn straight ...