Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
21 questions from the last 30 days
5
votes
1
answer
282
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An intriguingly simple integral functional for star-shaped, planar, simple, closed, smooth curves
The following is a question that popped up in my research in geometric analysis some time ago and that I dropped and kept coming back to multiple times. I will first state the problem, or rather my ...
- 159
4
votes
1
answer
183
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What is the volume of $\Sigma(2, 3, 13)$ associated with its $\widetilde{\operatorname{SL}(2, \mathbb{R})}$ geometry?
I've been considering a research topic based on extending the material from Khoi's research paper concerning a Chern–Simons-type invariant for 3-manifolds, and I'm stuck on a specific problem ...
4
votes
1
answer
201
views
Nonautonomous Lie derivative
Let $M$ be a smooth manifold with local coordinates $x = (x^1, \dots, x^m)$, and let $\tilde{M} = (t_1, t_2) \times M$.
Let a vector field $v(t, x) = (v^1, \dots, v^m)$ on $M$ depend on $t$ as a ...
- 175
10
votes
1
answer
325
views
Orientation local system of a vector bundle
Let $p:E\to X$ be a rank $k$ real vector bundle on a paracompact space. This question is about possible definitions of the orientation local system of $E$, which should be a local system of integer ...
- 37.6k
1
vote
0
answers
65
views
Heat kernel ratio on flat torus T² for specific winding sectors
Consider the heat kernel (Euclidean propagator) for a free particle on the space $T² × R³$, where $T²$ is a flat torus with radii $R₁$ and $R₂$.
The return amplitude for a path constrained to winding ...
- 11
6
votes
2
answers
555
views
When is a symplectic manifold with the opposite orientation itself symplectic?
Suppose $(M,\omega)$ is a closed (compact without boundary) symplectic manifold of dimension $2n$. Suppose $\overline{M}$ is a homeomorphic copy of $M$ with the opposite (reverse) orientation. My ...
- 693
2
votes
1
answer
300
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Topology of compact manifolds admitting codimension-one foliations with dense leaves
Let $M$ be a compact manifold endowed with a codimension-one smooth foliation $\mathcal{F}$, defined as the kernel of a closed, nowhere-vanishing 1-form $\omega \in \Omega^1(M)$.
It is classical that ...
- 41
1
vote
0
answers
251
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A question on rank two vector bundles on the real two sphere
Let $k$ be the real numbers, $K$ the compex numbers and let $x,y,z$ be independent variables over $k$. Let $f:=x^2+y^2+z^2-1$ and let $A:=k[x,y,z]/(f), B:=K[x,y,z]/(f)$ and let $S:=Spec(A)$.
In a ...
- 481
8
votes
0
answers
292
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Bottle-like Klein bottle
I spent a day trying to find a nice-looking bottle-like immersion of the Klein bottle.
I would like it to be analytic and reasonably simple, but all my attempts so far have produced rather ugly ...
- 47.3k
2
votes
0
answers
124
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Riemannian manifolds with maximal number of symmetries
This question is induced by what seems to be a rather large disconnect between "old" Riemannian geometry and modern treatments of it. For example Killing vector fields are extremely ...
- 3,685
3
votes
1
answer
106
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Ground eigenvalue of twisted Laplacian on constant-curvature Möbius strip with fixed embedding length
Consider a Möbius strip realized as a flat strip of length $L$ with identification $(y+L, w) \sim (y, -w)$. The twisted Laplacian acts on sections of the orientation line bundle; equivalently, on ...
- 41
5
votes
1
answer
221
views
Determinant line and directed sum
I'm reading Section 20.2 of "Monopoles and Three-Manifolds" written by P. Kronheimer and T. Mrowka. This section introduces the determinant line bundle $\det(P_s)$ of Fredholm operators $P_s$...
- 51
-5
votes
0
answers
118
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Spectral stability and non-orientable fibers in the inverse limit of a 2-degree covering map on the solenoid
I am investigating the dynamical properties of the solenoid $\mathbb{S}_{\mathbb{Q}} = \mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$, focusing on the spectral rigidity of its associated Hilbert space.In a ...
10
votes
1
answer
643
views
Formula for Riemann curvature tensor -- does it have a name?
Say you have $N$, an n-dimensional submanifold of a Euclidean space $\mathbb R^k$. We consider it to be a Riemann manifold with the pull-back metric. Locally near a point $p \in N$ you express $N$ ...
- 46.1k
2
votes
0
answers
60
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On the icosahedral projection of the $D_5$ root system and vertex figure decomposition
I am investigating the projection of the $D_5$ root system ($40$ roots) into $\mathbb{R}^3$ via the standard non-crystallographic $H_3$ embedding. Specifically, I am using the $3 \times 5$ projection ...
- 121