Relationship parallel transport, covariant derivatives and metrics
What follows is a discussion in "low level". For "high level" discussion go to connections summary.
- In an abstract setting of a
-dimensional manifold , given a parallel transport we can construct a covariant derivative operator . See the sketch of the construction at parallel transport#Covariant derivative, which can be generalized to any manifold, not only surfaces. - Conversely, given a covariant derivative operator
we can construct a parallel transport . See this construction. - A Riemannian metric
on gives rises to several covariant derivatives (or parallel transports). There is only one which is torsion free, the Levi-Civita connection , with parallel transport . - In
the standard metric gives rise to the standard covariant derivative which corresponds to a parallel transport , which is the usual absolute parallelism of . - If the manifold
is a surface immersed in (and I guess this is valid for any manifold immersed in ) we can construct a metric induced by the "ambient metric", . This inherited metric gives rise, according to point 3 above, to and . - In particular,
can be constructed from by means of geodesics and holonomy. See parallel transport#Intrinsic construction for surfaces. For 3-manifolds @needham2021visual provides three constructions in page 282. - But
can be constructed directly by means of translation and projection in , that is, by using and . This correspond to parallel transport#Extrinsic construction. - Respect to
, it can be constructed from , according to point 1 above, but also from and (see linear connection#How to get one).
What about the curvatures of the immersed submanifold and its relation the one of the ambient manifold? See Gauss' Equation.