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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?
Apr
10
comment How do I understand constraints on high order derivatives of the Gauss Map?
@Ben I added a few additional points to my answer. BTW, could you please uncover somehow, why would you sum these terms?
Apr
10
revised How do I understand constraints on high order derivatives of the Gauss Map?
added 2241 characters in body
Apr
10
revised How do I understand constraints on high order derivatives of the Gauss Map?
Fixed a bad typo
Apr
10
revised How do I understand constraints on high order derivatives of the Gauss Map?
edited body
Apr
10
answered How do I understand constraints on high order derivatives of the Gauss Map?
Mar
25
comment Flows of $f$-related vector fields
Try to use the uniqueness of solutions of ODEs
Mar
22
comment LU Factorization - Linear Algebra
@RedneckBlueState I think you've done pretty well. Errors is what we all learn from ;-)
Mar
22
comment LU Factorization - Linear Algebra
+1 for being faster that me :-)
Mar
22
answered LU Factorization - Linear Algebra
Mar
20
comment Germ induced by a submanifold
The submanifold $W$ passes through $p$ and defines the class $W_p$, so $W$ serves as a representative in $W_p$. In other words, this is the usual practice to define an equivalence relation $\sim$ on a set $A$, then pick an element $a \in A$ and refer to the class $[a]$ in $A/\sim$ as to induced by $a$. Of course, $W_p$ depends on $W$ in a sense (more precisely, determined by $W$), and for different $p$-s we get different classes $W_p$.
Mar
19
comment Germ induced by a submanifold
There is a definition of the germ of a submanifold here, which may help.
Mar
12
comment Deriving Ricci identity for co-vector fields
This is a widely used convention to assume that $\nabla$-s act on everything to the right, unless brackets are used. Also, there are pitfalls in this notation. I implicitly use the abstract index notation, which graphically is identical to the usual component notation, but really something like $2 \nabla_{[a} \nabla_{b]} X^c$ means a single tensor, more precisely a tensor field, a section of $T^* M \otimes T^* M \otimes T M$.