Book on differential Geometry with application to General Relativity

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Does anybody know of a good book on differential geometry that has applications to general relativity and also focuses on geometrical intuition?
I need a book that is not as rigorous as one that is targeted for mathematicians but also is not as loose as many books that are intended for physicists(although there are books for physicists that are rigorous enough).
Thanks!

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Based on the statements in the question I would recommend O'Neil's Semi-Riemannian Geometry with Applications to Relativity. O'Neil goes through manifolds and tensors with extreme rigor in the first 4 chapters, but later moves to a viewpoint more suitable for a physicist, so if you have tensor calculus under control you should be able to skip to the middle and get what you're looking for there.