Keith Burns & Marian Gidea 
Differential Geometry and Topology 
With a View to Dynamical Systems

Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer”s fixed point theorem, Morse Theory, and the geodesic flow.
Smooth manifolds, Riemannian metrics