Say we have a curve embedded in a 2-D manifold. I actually have two
notions of curvature for this curve (not counting intrinsic since it's
zero). One is its extrinsic curvature and the other its 4-
acceleration. It seems to me they must be somehow related but I have
no clue how or what the equation relating them is (I'm aware that one
is a tensor and the other a vector). Also, what about higher
dimensional embeddings? Can the extrinsic curvature be related to 4-
acceleration of curves that generate the embedding?

Thanks for any help!

On Feb 28, 10:17*pm, wrote:
Say we have a curve embedded in a 2-D manifold. I actually have two
notions of curvature for this curve (not counting intrinsic since it's
zero). One is its extrinsic curvature and the other its 4-
acceleration. It seems to me they must be somehow related but I have
no clue how or what the equation relating them is (I'm aware that one
is a tensor and the other a vector). Also, what about higher
dimensional embeddings? Can the extrinsic curvature be related to 4-
acceleration of curves that generate the embedding?

Thanks for any help!

I guess the trace of extrinsic curvature of the curve is proportional
to the projection of its 4-acceleration onto its normal (I'm using the
definition of the extrinsic curvature that assumes that the normal
vector is tangent to a geodesic).