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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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.
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.