This volume introduces techniques and theorems of Riemannian geometry,
and opens the way to advanced topics. The text combines the geometric
parts of Riemannian geometry with analytic aspects of the theory, and
reviews recent research. The updated second edition includes a new
coordinate-free formula that is easily remembered (the Koszul formula in
disguise); an expanded number of coordinate calculations of connection
and curvature; general fomulas for curvature on Lie Groups and
submersions; variational calculus integrated into the text, allowing for
an early treatment of the Sphere theorem using a forgotten proof by
Berger; recent results regarding manifolds with positive curvature.

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