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I am studying differential geometry but am having a hard time picturing curvature. Can anyone explain it to me in simple terms, perhaps with any diagrams. As simple as possible!

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You should probably be a little more specific as you can speak of the curvature of many different objects (e.g. curvature of a curve, curvature of a metric, curvature of a connection, etc.) I'm guessing you're asking about curvature of a Riemannian metric? – Eric O. Korman Oct 11 '13 at 17:06
It's the rate of change of direction with respect to distance. See my answer below. – Michael Hardy Oct 11 '13 at 20:14
I'm tentatively voting not to close because while it is difficult to give a precise answer to this question, I do believe that the intent of the question is relatively clear: "How to think about curvature?" So, I do feel it is possible to give a decent answer to this question. @Lucy Do you know the definition of curvature and, if so, with which definitions are you familiar? If you elaborate, then the chances of there being a precise answer to your question will improve. – Amitesh Datta Oct 12 '13 at 2:18
I have a couple of definitions; k=|T'|/|(gamma)'| where T=gamma'/|gamma'| – Lucy Oct 19 '13 at 21:14
As well as k=|gamma double dot| where dot=d/ds and k=theta dot where |theta dot|= |gamma double dot|. I also have definitions for normal curvature and geodesic curvature. – Lucy Oct 19 '13 at 21:19

The curvature of a circle is measured by the radius - the smaller the radius, the 'more curved' the circle is. This lets you define the curvature of any 1-dimensional manifold, since locally you can just 'fit a circle' to it. In the figure, the curvature of the manifold $\Sigma$ is greater at $P$ than at $Q$ because $r_1<r_2$.

I won't attempt further diagrams, but now think in 2-dimensions - you will need two different sized circles at every point in order to describe the curvature of the surface at that point. So you will have two curvatures (the principal curvatures), $\kappa_1$ and $\kappa_2$ which encode the same information as the sizes of the circles (if memory severs, $\kappa=1/r$ and $\kappa_1$ is the largest possible curvature and $\kappa_2$ is the smallest). Then the Gaussian curvature is the product of these, $$K=\kappa_1\kappa_2$$

In higher dimensions such simple descriptions don't work as well, and it's better to think about measuring curvature as what happens when you move around vectors. However, that concept follows from appropriate generalizations of the above concept.

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Curvature is how fast the direction is changing as a point moves along a curve.

In physical space, curvature is measured in radians per meter or radians per mile or degrees per mile, or the like. If you move one meter along a path, by how many degrees does your direction change? Divide the change in direction, in degrees, by the distance, in meters, and you've got the average curvature, in degrees per meter. Then take the limit as that one-meter-long curve shrinks to a point, and you've got the curvature in degrees per meter at that point.

In pure mathematics, one often regards distances as mere numbers instead of multiplying a number by a unit of measurement. Then the curvature becomes a dimensionless number.

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