curves' curvature is simply radius of osculating circle inverted. A surfaces' curvature varies with orientation of plane that cuts the surface into curve-but at two orientations, curvature reaches extremum. location of extremum as well as average of the curvatures do not change with orientation. Thus we have Gaussian (and Mean) curvature of surface embedded in 3D space. What about 3D hypersurface embedded in 4D space? (i.e. motivation for generalization behind). Do we cut the hypersurface with hyperplane and find average gaussian curvature of the generated 2D surfaces?
The idea behind generalised curvatures is pretty non-trivial, but the basic principle is something like this: imagine being on a surface, facing some direction. Now move around in a circle, facing a the same direction as you did before. If you move around are region with non-zero curvature, when you get back to the start, you'll end up facing a different direction to the one you started with. A good demonstration of this can be done with a cone of paper, draw a circular path around it, then open it up and draw parallel lines on the path, then join it back together and you'll find that the parallel lines are no longer so at the join. The geometric object that describes this in the general case with this is called the Riemann Curvature Tensor, a mathematical object that describes the amount of change in the direction of a vector as it goes around a really small closed loop. This can then be summarised into other quantities such as the Ricci tensor, and the scalar curvature which in 3D is equal to two times the Gaussian curvature.
It lets you know something about the energy of a n-dimensional object.
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