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Results tagged with dg.differential-geometry
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user 259525
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8
votes
0
answers
81
views
Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there a …
- 1,429
8
votes
1
answer
808
views
General wedge-product for vector bundle valued forms
In mathematics and physics, especially gauge theory, there are many different but related notions of wedge products when discussing vector space- and vector bundle-valued differential forms. For examp …
- 1,429
6
votes
1
answer
180
views
On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (co …
- 1,429
5
votes
1
answer
352
views
Spin connection in the tetradic Palatini-formalism of general relativity
$\DeclareMathOperator\SO{SO}$I am trying to understand the tetradic Palatini-formalism of general relativity from a mathematical point of view. I am graduate student and quite new to mathematical gaug …
- 1,429
5
votes
1
answer
245
views
Space of spacelike embeddings as infinite-dimensional manifold
Consider a four-dimensional Lorentzian manifold $(\mathcal{M},g)$ and a $3$-dimensional compact manifold $\Sigma$, such that there exists a spacelike embedding $i:\Sigma\to\mathcal{M}$ so that $h:=i^{ …
- 1,429
4
votes
1
answer
420
views
Etymology “Kulkarni–Nomizu product”
$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two symme …
- 1,429
3
votes
0
answers
121
views
On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one rec …
- 1,429
2
votes
0
answers
130
views
Definition of trace in topological BF-theories
I very important example of topological field theories are "BF-theories", which are usually defined as follows: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\mathcal{M}$ be …
- 1,429
2
votes
0
answers
125
views
Elliptic equations and Fredholms alternative in the non-compact case
Let $M$ be a smooth Riemannian manifold and $E$ be a finite-rank vector bundle over $M$ equipped with a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$, i.e. $\la …
- 1,429
2
votes
2
answers
136
views
Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?
Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that
$$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$
where $\Sigma …
- 1,429