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Consider a smooth surface for simplicity. What does its curvature measure? What does its Gaussian/Riemannian curvature measure? What does its torsion measure?

What does the Ricci curvature measure?

Cameron Buie
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superAnnoyingUser
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    I think you can get all them in any textbook on differential geometry. You shall specify your problem. – Shuchang Sep 13 '13 at 07:19
  • None of the textbooks I read deal with examples or interpretation of the concepts, sir. And I shall not specify any further, as this question stems from the most general discussion. @ShuchangZhang – superAnnoyingUser Sep 13 '13 at 08:13
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    The word "curvature" is vague: there are at least 10 different notions of curvature. The Gaussian curvature and Riemann curvature tensor are different things. The word "torsion" also has two different meanings -- namely (1) the "torsion tensor" that measures the rotation of vectors along geodesics, and (2) the torsion of an embedded curve in $\mathbb{R}^n$, which measures its deviation from being planar. – Jesse Madnick Sep 13 '13 at 08:43
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    See this answer for geometric interpretations of the curvature and torsion. – Sepideh Bakhoda Sep 14 '13 at 12:50
  • If the best answer is just a link to a previous question, that means the question itself is a duplicate of that one, doesn't it? –  Sep 22 '13 at 03:55

2 Answers2

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If you search the questions on MSE and/or MO, I think you will,find some pretty good insights into these topics; if I recall correctly, these topics have come up more than once; many times, in fact. For example, this one might be a good place to start:

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

Good luck in you searches!

Community
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Robert Lewis
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The curvature literally measure how sharp a curve is at a certain point. For example if you have a circle of small radius then the curvature (which is given by $\frac{1}{R}$ for a circle) at any given point will have a large curvature which means that it is bending sharply. Similarly, if you have a large radius then the curvature will be much smaller which means that it is bending less sharply. The Torsion of a curve measure how sharply it is twisting out of its plane curvature. I perfect example to look at would be the helix. (To be specific when I talk about a small radius I am referring to $R<1$.)

The Ricci curvature tensor provides a way to which the geometry provided by the Riemannian metric might differ from ordinary Euclidean n-space. An application of the Ricci curvature tensor is in General Relativity in connection with the Einstein Field Equations.

RDizzl3
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