Relationship parallel transport, covariant derivatives and metrics

What follows is a discussion in "low level". For "high level" discussion go to connections summary.

  1. In an abstract setting of a n-dimensional manifold M, given a parallel transport τ 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.
  2. Conversely, given a covariant derivative operator we can construct a parallel transport τ. See this construction.
  3. A Riemannian metric g on M gives rises to several covariant derivatives (or parallel transports). There is only one which is torsion free, the Levi-Civita connection g, with parallel transport τg.
  4. In RN the standard metric gstd gives rise to the standard covariant derivative std which corresponds to a parallel transport τstd, which is the usual absolute parallelism of R3.
  5. If the manifold M is a surface immersed in R3 (and I guess this is valid for any manifold immersed in RN) we can construct a metric induced by the "ambient metric", gind. This inherited metric gives rise, according to point 3 above, to ind and τind.
  6. In particular, τind can be constructed from gind by means of geodesics and holonomy. See parallel transport#Intrinsic construction for surfaces. For 3-manifolds @needham2021visual provides three constructions in page 282.
  7. But τind can be constructed directly by means of translation and projection in RN, that is, by using τstd and gstd. This correspond to parallel transport#Extrinsic construction.
  8. Respect to ind, it can be constructed from τind, according to point 1 above, but also from std and gstd (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.