Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,777
questions
1
vote
0
answers
14
views
Characterization of self-conjugate spin$^c$ structures
Let $M$ be an oriented Riemannian $n$-manifold. Then we can choose a trivializing open cover $M=\bigcup_\alpha U_\alpha$ for $TM$ and corresponding transition functions $g_{\alpha \beta}:U_\alpha \...
- 249
1
vote
0
answers
36
views
Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
- 500
3
votes
0
answers
74
views
Decomposition of forms in $\operatorname{SU}(4)$-manifold
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I ...
- 923
6
votes
1
answer
153
views
Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
- 3,164
4
votes
1
answer
134
views
Lie algebra cohomology and Lie groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}\DeclareMathOperator\Sp{Sp}$Let $G$ be a complex Lie group and $\mathfrak{g}$ its Lie algebra (which is over $\mathbb{C}$). My first question is:
(...
- 627
1
vote
0
answers
82
views
Dirac operator on $\operatorname{Spin}(7)$, $G_2$ and $\operatorname{SU}(3)$ manifolds
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to ...
- 923
1
vote
0
answers
140
views
measure of Haar
Let $(G,K)$ be a Gelfand pair.
Why, for a function $f$ $K$-binvariant with respect to a compact subgroup $K$ of a group $G$, do we have the following equality:
$$ f(xy) = \int_K f(xky) \, dk$$
A ...
- 29
2
votes
0
answers
32
views
Bounding norms of symplectic matrix factorisations and non-separable Hamiltonian flows
Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc}
A & C\\
C^T & B\\
\end{array}\right)$ ...
- 21
1
vote
0
answers
38
views
Making sense of constant scalar curvature metric horns
Suppose we have a compact oriented surface $S$ and we remove a point $p$ on it. We could consider a neighboorhood $U$ of the puncture $p$, so that the points in this neighboorhood are described by ...
2
votes
1
answer
102
views
Slowly increasing smooth mappings with values in a Lie group?
Let $G$ be $l$-dimensional compact Lie group and consider any smooth $F : \mathbb{R}^n \to G$.
Then, the first-order derivative of $F$ at each $x \in \mathbb{R}^n$ can be regarded as a linear mapping $...
- 3,113
9
votes
1
answer
516
views
Citation hunting: Floer on spectral sequences
I vaguely remember a YouTube talk that began with a citation from Floer regarding the existence of a spectral sequence. The idea was that given a manifold with a Morse function, we can construct a ...
- 153
9
votes
3
answers
682
views
A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \...
1
vote
0
answers
59
views
Applicability of van Holten's algorithm for symmetries in classical mechanics
Background
van Holten's algorithm (see e.g. here and here) is a way of constructing or recognizing dynamical/hidden symmetries in classical mechanics by looking for Killing tensors on the ...
- 119
0
votes
1
answer
82
views
Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces
We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...
- 101
1
vote
0
answers
38
views
Uniform bound for coefficients of fundamental forms of unbounded surface
Recently I am using the Gauss-Weingarten relations to transform an equation defined on a domain $\Omega\in\mathbb{R}^3$ into a form expressed in local coordinates when it approaches the boundary $\...
- 71