1. What is the metric tensor?

  2. How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices?

  3. How it relates to distance function (metric) and angles?

  4. How does it transport basis vectors from one coordinate system to another?

  5. How is it different from the field tensor, Riemann curvature tensor and Ricci curvature tensor?

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    $\begingroup$ Have you looked at a book? Seriously there are many comprehensive books and websites on this subject. Googling for "mathematical foundations of general relativity" and "differential geometry" will go a long way. $\endgroup$ – Michael Brown Aug 9 '13 at 7:08
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    $\begingroup$ I also suggest Ch 14, called "Calculus on Manifolds" of Roger Penrose's "The Road to Reality" as a pithy summary of the foundations of Riemannian geometry. He does an awful lot with very little, gives a good explanation of the notions of curvature before introducing the metric. Exercises in the book are well worth doing, and there is a website where readers have put their solutions. One I also like for its pithiness, thoroughness and clarity is Wulf Rossmann's "Differential Geometry" - and its free! $\endgroup$ – Selene Routley Aug 9 '13 at 7:30
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    $\begingroup$ It's a bit funny to see a question that says "What is X?" with a wp link on X. $\endgroup$ – Abhimanyu Pallavi Sudhir Aug 9 '13 at 7:39
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    $\begingroup$ You might want to break this up into several questions. $\endgroup$ – Dan Aug 9 '13 at 8:16
  • $\begingroup$ "Road to Reality" is great if you want a heavy book on just about everything foundational in modern physics. If you would like a modern, breezy, "physics first" approach devoted to general relativity with plenty of interesting historical footnotes I'd recommend Zee's book "Einstein Gravity in a Nutshell". Then there is the classic tome by Misner, Thorne and Wheeler. The advanced theory and experimental sections are a little out of date, but the book is truly great if you want geometric intuition with pictures... oh so many pictures... $\endgroup$ – Michael Brown Aug 9 '13 at 8:51

The metric tensor is a quadratic form/rank-2 tensor that encodes the geometry of a manifold by $ds^2=g_{\mu\nu}dx^\mu dx^\nu=dx^\mu dx_\nu$. Another way the definition is often stated is that the metric allows you to convert between contravariant and covariant indices, e.g. vectors to covectors as $g_{\mu\nu}x^\mu=x_\nu$, or equivalently that it allows you to take the dot product between vectors (because $x^\mu y_\mu$ is invariant, while $x^\mu y^\mu$, if you computed it, would not be -- this is why repeated indices always appear top and bottom to cancel out in relativity).

The whole point of the metric tensor is that the mixed form (vector-to-vector transformation form), $g^\mu{}_\nu$, is just the identity. This is unlike transforming covectors to vectors, etc. where the representations can be dfiferent.

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