TITLE: What is the difference between intrinsic and extrinsic curvature?
QUESTION [10 upvotes]: In general relativity, energy bends spacetime. However, this doesn't mean that a fifth dimension for spacetime to "bend into" exists." That is, spacetime isn't embedded in a higher dimensional space, Instead, the curvature is said to be intrinsic.
But what does that mean? One could imagine the sphere on a ball as an example of extrinsic curvature it seems that intrinsic curvature isn't as straight-forward and intuitive. Is there a simple and easy way to understand the differences, and how a space can "curve" without actually be embedded in a higher dimensional space to "bend" in?

REPLY [7 votes]: A Riemannian manifold is intrinsically curved if there exists a geodesic triangle bounding a topological disk whose interior angles do not add to $\pi$.
An embedded Riemannian submanifold $M \subset N$ is extrinsically curved if no smooth orthonormal frame of normal vectors of $M$ is parallel (convariantly constant) in $N$.

A flat square torus embeds isometrically as a product of circles in four-dimensional space; the image is intrinsically flat (every triangle bounding a disk has total interior angle $\pi$) but extrinsically curved (there is no parallel field of unit normal vectors, much less a parallel frame).
A great sphere in a round $3$-sphere is intrinsically curved (isometric to the unit sphere in Euclidean $3$-space, so a triangle bounding a disk has total interior angle $> \pi$) but extrinsically flat (each of the two smooth unit normal fields is parallel). If you don't mind a non-compact example, a circular cylinder in Euclidean $3$-space is intrinsically flat but extrinsically curved for similar reasons.

Gravitational lensing may be viewed as a physical manifestation of intrinsic curvature: In an approximation where the lensing system is static, the path of a light ray is a spatial geodesic. Two light rays seen at distinct points of the sky form a "digon", a geodesic polygon with two sides. Because the sum of the interior angles is not zero, space is intrinsically curved.