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May
11
revised Curvature tensors and bivectors
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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
May
4
answered Dual tensor for partial derivative, if it has any meaning
Apr
28
revised What is a local invariant?
added 4 characters in body
Apr
28
answered What is a local invariant?
Apr
28
answered When is the pullback of a tangent bundle along a curve a tangent bundle on the curve?
Apr
28
comment When is the pullback of a tangent bundle along a curve a tangent bundle on the curve?
@Mike There are many various non-isomorphic bundles on a curve, say trivial bundles $I \times \mathbb{R}^k$ of the ranks $k = 1,2,\dots$. In fact, since the segment $I$ is contractible, there are only trivial bundles on $I$. I guess this is what you mean.
Apr
20
comment Induced Connection on $\Sigma\subset M$
The best known to me reference on this topic is B. Andrews, C. Hopper, The Ricci Flow in Riemannian Geometry, see section "1.8 Pullback Bundle Structure" on pp.24-27 with all the proofs. The book is available online on the 1st author's webpage.
Apr
14
revised Curve Orientation on a Surface
added 251 characters in body
Apr
14
answered Curve Orientation on a Surface
Apr
12
revised How many degrees of freedom are in a flat metric and how does one count them?
added 1 character in body
Apr
12
answered How many degrees of freedom are in a flat metric and how does one count them?
Apr
12
comment How do I understand constraints on high order derivatives of the Gauss Map?
Of course, $T^(i,j)$ is a complete contraction in a sense, but I wanted to say that we take the dot-product of two vectors, whereas the tensors $D^i N$ are not contracted between themselves in the covariant slots. May be, I should have just said that we simply multiply two vectors by the Euclidean dot product in the ambient space, and speaking about "complete contractions" is an overkill.
Apr
12
comment How do I understand constraints on high order derivatives of the Gauss Map?
@Ben I am trying to understand a similar analytical technique but for a hypersurface in an arbitrary Riemannian manifold, an analogue of Taylor theorem, so I am quite interested in this discussion. I have been working in conformal geometry of hypersurfaces for the last few years. It is perhaps too early to announce the results and conjectures publicly, but if you wish we can communicate privately.
Apr
11
answered How do I understand constraints on high order derivatives of the Gauss Map?