Yuri Vyatkin
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 Mar 21 revised Affine connection, metric and parallel transport and mutual interdependence Fixed a typo in the main formula Mar 20 answered Affine connection, metric and parallel transport and mutual interdependence Mar 13 comment Exterior Derivative vs. Covariant Derivative vs. Lie Derivative @AlexM. No, I would not say that. As I mentioned in the beginning, connections involve choices (and are abundant). Of course, the naturality of the Levi-Civita connection (which is actually is treated in the reference to (3)) is the best illustration of what I was intending to allude. Another good example would be the naturality of the pullback connection. Giving precise statements was not what I was aiming at in this answer, but I tried to provide references for the curious to dig further. Mar 4 comment The space of Riemannian metrics on a given manifold. @AlexM. You have spotted a sloppy point. Thanks for that. I should have said something like "locally like a Fréchet space". For the details please refer to e.g. David G. Ebin, On the space of Riemannian metrics Feb 14 comment Is every scalar differential operator on $(M,g)$ that commutes with isometries a polynomial of the Laplacian? Thanks, @Bombyxmori Feb 14 awarded Nice Answer Dec 6 awarded Good Answer Sep 20 awarded Yearling Jul 28 comment How do I understand constraints on high order derivatives of the Gauss Map? @Ben I replied to you via e-mail, you can delete the above comment now to avoid spammers. Jul 8 awarded Good Answer May 27 revised Need help understanding a relation between the fundamental forms Corrected some minor typos and omissions May 27 revised Need help understanding a relation between the fundamental forms added 2885 characters in body May 27 answered Need help understanding a relation between the fundamental forms May 11 revised Curvature tensors and bivectors added 2 characters in body May 11 answered Curvature tensors and bivectors May 6 comment Index notation. @bosco yes, this way of using semicolon is just another notation for the covariant derivative: $T_{j k \dots}{}^{l m \dots}{}_{;i} \equiv \nabla_i T_{j k \dots}{}^{l m \dots}$. May 6 answered Index notation. May 4 comment Dual tensor for partial derivative, if it has any meaning @Valery You are welcome. If you mean to strengthen your (multi)linear algebra, Sergei Winitzki, Linear Algebra via Exterior product is highly recommended. May 4 revised Dual tensor for partial derivative, if it has any meaning added 11 characters in body May 4 revised Dual tensor for partial derivative, if it has any meaning added 12 characters in body