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I have started an attempt to self-study Riemannian Geometry.
I well understand all the algebraic properties of the Riemann tensor (With a symmetric connection), and how it gradually becomes less and less frightening.

However, I'm having trouble converting all the properties of this tensor into an explanation of how it represents curvature, and the same goes for the derived Ricci tensor. This is essentially what I'm aiming for in writing this question.

I can decompose this question into a set of concrete questions, but I think it is too "early" to ask them with regards to my understanding of the subject.

A Reference to relevant material would suffice.

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    $\begingroup$ Just read up. For instance, this survey gives a pretty modern yet accessible perspective. $\endgroup$ – Yuri Vyatkin Jan 22 '14 at 8:26
  • $\begingroup$ I couldn't quite find an applet that would do what I wanted, although things like torus.math.uiuc.edu/jms/java/dragsphere were close. Basically, I would recommend doing an explicit approximate calculation of the Riemann tensor (and then sectional curvature) for a sphere with the two directions starting out orthogonal. Only from working through the definitions with a toy case like that did I get the sense that "positive Gaussian curvature really should have positive sectional curvature as defined in terms of the Riemann tensor", which really helped me feel better about it. $\endgroup$ – Mark S. Mar 9 '14 at 21:16

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