  These notes on Riemannian geometry use the bases bundle and frame bundle, as
in Geometry of Manifolds, to express the geometric structures. It has more
problems and omits the background material. It starts with the definition of
Riemannian and semi-Riemannian structures on manifolds. Affine connections,
geodesics, torsion and curvature, the exponential map, and the Riemannian
connection follow quickly. The Taylor series for of the metric in normal
coordinates is an unusual feature. The last third covers, first and second
variation of energy, completeness, cut points, the Hadamard-Cartan theorem, and
finishes with the use of matrix Riccati equations to prove the volume
comparison theorem. There is a comprehensive index.
