I want to understand the basic conceptual idea about intrinsic and extrinsic curvature.
If we consider a plane sheet of paper (whose intrinsic curvature is zero) rolled into a cylindrical shape, then we say that its extrinsic curvature is non-zero. So how can I visualize the extrinsic curvature?
I read somewhere that the extrinsic curvature indicates how the 2D surface is embedded in 3D space. So what does it mean?
[Math] Intrinsic and Extrinsic curvature
curvaturedifferential-geometry
Best Answer
Explaining the idea of a 2 D surface embedded in 3 D Space.
You will get a (much) better answer than this, but a straightforward reason is that for using extrinsic curvature, we need an extra dimension to put the lower dimensional object "into".
Intrinsic curvature of a surface or manifold can be performed by using math techniques developed by Gauss and Riemann and allows us to do without the extra dimension in curvature calculations.
Obviously everything we see around us is actually in 3D, but we can pretend a sheet of paper is 2D, so looking at a sheet of paper rolled up into a cylinder is looking at a 2D surface in a 3 D space and we can measure directly how much it's curved.
Image Source: Descriptions of Curvature
From looking at these 3 shapes, it's quite obvious that in a 3 D world, we could physically measure all aspects of them, and determine which of them has, in any small region on it, positive curvature, (the sphere), negative curvature, (the "vase" shape) and which has no curvature, (the middle cylinder).
So 2D embedded objects in 3D space are easy to measure.
But we have to describe 4 D objects and events in a 4 D space, so we don't have the "elbow room" of an extra dimension.
So we give up on the visual attempt ( which is why most, if at all depictions of real 4D on TV and books are misleading) and we use expressions such as this:
$${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$
to calculate how much spacetime is curved.
This above is an example using intrinsic curvature, we don't (and actually can't) "lift " ourselves above the space.
We are in 4 D space-time, so to perform calculations in this space, we must use intrinsic curvature math ( differential geometry) as we can't go to a higher dimension.
In slightly more specific terms,you might have read about the metric tensor, which is based around the idea of using basis vector and their components to calculate distances (in 4 dimensions) utilising differential geometry to measure deviations from flat space caused by massive objects.
The other benefit of the intrinsic based metric tensor is that it is applicable in all GR calculations, (invariance).