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After some time of studying differential geometry/topology while avoiding riemmanian geometry I find mysef at a wierd position. I'd like to read a book that contains an introduction to riemannian geometry that assumes familiarity with basic differential geometry/topology (transversality and degree, vector and tensor bundles, differential forms and integration, lie groups etc.). But instead of it going deaper into rimannian geometry (comparison theorem and other geometric analysis stuff) i'd like it to cover some advanced matirial on differential geometry as well. Ideally it would contain a (not too brief) introduction to these concepts:

  • Basics of riemannian geometry (affine connection, metric tensor, gauss bonnet, curvature forms and tensors, jacobi fields)
  • Connection and curvature on fibre bundles
  • Holonomy and folliations
  • Jet bundles
  • Differential operators (Laplacian, and Dirac)

Is there such a book? Is it too much to ask for in one book?

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Aside from jet bundles and connections on general fiber bundles, I believe Peter Petersen's Riemannian Geometry has everything you ask for. You can get it here.

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Let me answer your last questions first: "Is there such a book? Is it too much to ask for in one book?" Yes, indeed; I know of no book covering all the subjects you are asking for.

A good book with some of the basics you want to learn about Riemannian Geometry is Jost, Riemannian Geometry and Geometric Analysis. In addition, it contains a nice chapter on "Morse Theory and Floer Homology" (a subject normally found in books on differential topology). Another good introduction, with two chapters on fiber bundles and characteristic classes, is Morita's Geometry of Differential Forms (with solutions to the exercises). Finally, for the other topics you mention, I think Kolar, Michor and Slovak, Natural Operations in Differential Geometry may be of great help to you.

Hope it helps.

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