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Aug
18
comment Understanding Penrose diagrammatical notation
the juxtaposition of "coupons" seems to be reserved to the composition of morphisms A covariant (or mixed) tensor is a linear transformation, which is a morphism. For example, $M^a_b N^b_c$ can be interpreted as a notation for matrix multiplication, which is the composition of two morphisms. contraction contrarily implicates some sort of Einstein summation, am I right? Penrose's birdtracks notation is isomorphic to abstract index notation. Indices don't take integer values or imply a choice of basis. Summation only happens if you convert to concrete index notation.
Aug
17
comment The momentum constraints in the ADM formulation of general relativity
There is a standard way of extending the definition of the covariant derivative to include tensor densities; see the end of this section of the WP article: en.wikipedia.org/wiki/… . They actually address the derivative we're talking about as an example of a tensor density. Maybe another way of getting at it is the following. I assume that this generalization obeys the product rule, obeys the chain rule, and is metric compatible. So the derivative we're talking about can be evaluated down to derivatives of components of g, and these vanish.
Aug
17
comment The momentum constraints in the ADM formulation of general relativity
I think the answer is right, but the comment is not. In the case of a (0,0) tensor, the covariant derivative is the same as an ordinary partial derivative. Therefore if the square root of the determinant of g were a (0,0) tensor, then its covariant derivative would be the same as its ordinary partial derivative, which does not in general vanish.
Aug
8
comment Mathematical software wish list
My wish would be for open-source software that can plot surfaces with realistic shading.
Aug
4
comment How much of differential geometry can be developed entirely without atlases?
related: math.stackexchange.com/questions/53021/…
Aug
4
comment How is differential geometry used in immediate industrial applications and what are some source to know about it?
This is just a vague memory, but I think there may be applications in fields like robotics, where the robot has various degrees of freedom (angle of this joint, length of this telescoping part, ...), and the space describing these degrees of freedom is a manifold.
Jul
29
comment Distributing points evenly on a sphere
What's most surprising to me is that there is a link between a global property extrinsic to the surface (energy) and a local, intrinsic property (curvature).
Jul
29
awarded  Pundit
Jul
29
comment Distributing points evenly on a sphere
There is a problem similar to the Thompson problem, which is to find the minimum potential energy of a continuous charge distribution on a nonspherical conductor. There is a surprising set of exact solutions for some special shapes, in which the charge density is proportional to the fourth root of the absolute value of the Gaussian curvature. I W McAllister 1990 J. Phys. D: Appl. Phys. 23 359 doi:10.1088/0022-3727/23/3/016
Jul
26
awarded  Civic Duty
Jul
12
awarded  Notable Question
Jul
1
comment Ordinals separate from set theory
Conway's surreal numbers include the ordinals. They can be developed separately from set theory, and that's what Conway did in "On Numbers and Games." Some discussion here: personal.psu.edu/t20/fom/postings/9905/msg00074.html
Jun
22
comment Flat Riemannian manifold
This doesn't seem like a research-level question. It might have been more appropriate for math.SE.
Jun
22
comment Finding Riemannian metric for this geodesic
I don't understand the motivation for the question. Since no coordinate system has been specified, isn't it vacuous? For example, I could just associate the $tb^i$ with the Cartesian coordinates in a flat space, and take the given equation to be a change of coordinates. And what is the motivation for the constraint on the sum of the $\gamma$'s? This would simply seem to restrict our attention to a $(d-1)$-dimensional submanifold.
Jun
21
comment What notions are used but not clearly defined in modern mathematics?
This seems to me to be a poor answer, for the same reasons I gave in a comment on the answer by Buschi Sergio.
Jun
21
comment What notions are used but not clearly defined in modern mathematics?
This is very unclearly written and doesn't make much sense to me. For example, "Concept and existence of points, and spaces as sets of points." This sounds like a description of an axiomatic style of mathematics, in which we have certain primitive notions that we make no attempt to define explicitly. This is part of the normal process of mathematics and does not indicate that something is "not clearly defined" in the nontrivial sense of the OP's question, as I understand it.
May
19
comment Kinematics of rolling knots
The relevant material seems to be at the end of the talk, around 9:00? One thing to think about would be whether you want to ask these questions about knots with ideal shapes, or knots that are arbitrary fattened embeddings of the topological knot in 3-space. Ideal shapes are poorly characterized: arxiv.org/abs/1402.5760 . It's not obvious to me whether the dynamics are Hamiltonian. See, e.g., en.wikipedia.org/wiki/Chaplygin_sleigh .
May
19
comment Kinematics of rolling knots
possibly related: mathoverflow.net/questions/180194/…
May
19
comment Flat Riemannian manifold
An example that the OP might wish to think about would be a cylinder with a flat metric. That might help to clarify why the answer is framed in local terms.
May
18
awarded  Necromancer