What is the intuitive, geometrical meaning regarding the usual mixed Riemann tensor $R^{a}{}_{bcd}$ with respect to its purely covariant counterpart $R_{abcd}$?


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There is no deep intuitive geometrical meaning behind a Riemann tensor with some indices moved up/down. You could say that the two variants are "dual" to each other, loosely speaking. The only new information that arises from raising or lowering an index is the underlying metric tensor. So that's where your geometric interpretation can come in, if you want to think in that way.

As an example, consider the simplest vector you can think of: the infinitesimal position coordinates $dx^\mu$. What is the geometrical meaning of $dx_\mu$ compared to $dx^\mu$? They represent the same physics, but are still dual to each other. They are dual in the sense that when you combine the them and contract the indices, you get a scalar quantity. In special relativity, you will get $ \eta_{\mu\nu} dx^\mu dx^\nu = ds^2$, where $ds^2$ is a scalar quantity that represents the (square of) the proper time. The metric tensor $\eta_{\mu\nu}$ makes the inner product work.

This idea can be generalized to any tensor with any number and positioning of indices, like the Riemann tensor. So raising or lowering indices is, fundamentally, just another way of defining an inner product under the hood, and also defining some kind of scalar.

The important question, then, is does that scalar represent something physically/geometrically?

  1. For $dx^\mu$, the constructed scalar represented proper time.

  2. For momentum $p^\mu$, the constructed scalar represents the mass.

  3. For the metric tensor itself $g_{\mu \nu}$, the constructed scalar represents the number of spacetime dimensions.

  4. For the Riemann tensor, there are four scalars you can construct from a pair of Riemann tensors, three of them being $R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma}$, $R_{\mu \nu} R^{\mu \nu}$ and $R^2$. The first one is often used as a measure of curvature. Another interesting scalar that you can create from a combination of these is the Gauss-Bonnet term.


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