All Questions
Tagged with dg.differential-geometry connections
201 questions
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On the correspondence between infinitesimal and integral description of connections
It is the title of an article by Petko Nikolov Triste Sissa 1981. I cannot access this pdf yet I remember that it was once avaliable on libgen and now I cannot find it. Please help.
- 4,713
1
vote
1
answer
117
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Is every connection locally flat for an other connection?
Consider a $C^{\infty}$ connection $d_A = d+A$ on the unit ball $B^n\subset \mathbb{R}^n$. Does there exists another connection $d_{\tilde{A}} = d+\tilde{A}$ such that $d_{\tilde{A}} A = 0$? That is ...
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2
votes
1
answer
533
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Bianchi's identity in a principal bundle
Let us consider a principal bundle $P$, with a Lie-algebra-valued connection one-form $\omega\in\mathfrak{g}\otimes\Omega^1(P)$ and a Lie-algebra-valued curvature two-form $\Omega\in\mathfrak{g}\...
CommunityBot
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12
votes
3
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639
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Embedding of a bundle with connection into a bundle with flat connection?
I'm looking for a generalization of Nash's embedding theorem (for Riemannian manifolds) to vector bundles with a connection.
Given a smooth manifold $M$ together with a vector bundle $V$ on $M$ ...
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2
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1
answer
929
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Characterisation of (integrable) connections on (trivial) principal bundle
Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra.
Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq ...
CommunityBot
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0
votes
1
answer
255
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Torsion free Chern connections and Kähler manifolds
Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
CommunityBot
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3
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4
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3k
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Alternative (easier) Proof of Ambrose Singer Holonomy theorem
Let $P(M,G)$ be a principal bundle. Giving a connection on $P(M,G)$ means two equivalent things. One as an assignment of subspace of $T_pP$ for each $p\in P$ and another as a $\mathfrak{g}$ valued $1$ ...
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4
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0
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104
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Can we get a connection on the principal bundle from a connection on the associated vector bundle?
Assume $G$ is a Lie group, $P \to M$ is a smooth principal $G$-bundle, and $\rho \colon G \to GL(V)$ is a smooth representation of $G$. We can define a connection on the associated vector bundle $E := ...
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11
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3
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886
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Tangent bundle of a tensor product bundle
This question was also asked here on math-stackexchange.
Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K_E: ...
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3
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504
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Connection Transformation Formula; Degree 3 Cech Cohomology
While reading through Brylinski, as in all of my posts, I am trying to understand the following equation:
$ g_* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$
Setting
I have a principal $B$-bundle, $Q$...
- 1,109
3
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0
answers
89
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Torsion in terms of parallel transport
This MO answer establishes the connection between parallel transport and torsion in the special case of the canonical flat connections on a Lie group. This suggests a more general construction for an ...
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2
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0
answers
53
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Connection vs Exponential preserving maps
Connection Preserving Diffeomorphisms
The setting is a manifold $M$ equipped with a linear connection $\nabla$. Kobayashi & Nomizu [K&N §VI.1] define a connection preserving diffeomorphism (...
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5
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1
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Converging paths implies converging parallel transports along those paths?
Suppose we have a vector bundle $E$ with connection $\nabla$ over a smooth manifold $M$. Let’s also say we have a sequence of smooth paths $\gamma_n\in C^\infty([0,1],M)$ starting at the same point $\...
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Extending $G$-equivariant local diffeomorphisms on principal bundles to local bundle maps
Consider a principal $G$-bundle $P$ over the base space $M$ equipped with a connection 1-form $\omega$. Let $\mathcal{U}$ and $\mathcal{V}$ be open subsets of $P$, and suppose $F: \mathcal{U} \to \...
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0
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411
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Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates
Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle
$$TM \vert_{\...
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0
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163
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Parallel transport of global sections and Riemannian curvature
A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days.
Consider a (real) smooth ...
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8
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0
answers
318
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Flat Maurer-Cartan connection iff flat Berry connection
I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$.
The first is the canonical or $H$-...
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29
votes
4
answers
4k
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Rolling without slipping interpretation of torsion
This is, in a sense, a follow up to this question.
Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...
- 1,446
4
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1
answer
241
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Let $G \subset \mathrm{GL}(n)$ be a matrix Lie group. Does there exist an affine connection under which the matrix and manifold exponential coincide?
Let $G\subset \operatorname{GL}(n)$ be a matrix Lie group. I am curious about curves $\gamma(t) = g \exp(tv)$, where $g \in G$, $v \in \mathfrak{g}$, and $\exp(.)$ is the matrix exponential. If we ...
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6
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1
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273
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Commutative/ symmetric second covariant derivative
Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$.
Is it possible to have an affine connection, possibly with non-zero ...
- 108k
1
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0
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153
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Torsion free connection $\implies$ Jet coordinates $=$ Taylor expansion coefficients?
Suppose we have some smooth n-dimensional manifold $M$ endowed with basis 1-forms $\theta^a$ with $a=1\cdots n$. Then $\theta^a$ are sections of the coframe bundle $F^* M$. In local coordinates ($x^a$ ...
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1
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387
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Torsion-free Cartan connections
Let $M$ a differentiable manifold and $H\subset G$ a Lie group with a closed subgroup such that $G/H$ is connected. A $H\subset G$-Cartan connection on $M$ can be defined by
A principal $G$-bundle on ...
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1
vote
0
answers
155
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Lifting action of torus to torus bundle
Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...
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1
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1
answer
197
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An identity for the higher form Levi-Civita connection
Take $M$ a Riemannian manifold and $\Lambda^1$ its space of one forms. The LCC (Levi-Civita connection) $\nabla:\Lambda^1 \to \Lambda^1 \otimes \Lambda^1$ is well known to satisfy the identity $m \...
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4
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2
answers
726
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Obstructions to the existence of a flat connection on a vector bundle
Given a smooth manifold $M$ and a smooth vector bundle $E \to M$ (with real or complex fibers), what are known obstructions to the existence of a flat connection on $E \to M$? If all known ...
- 108k
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0
answers
169
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Understanding the Seiberg-Witten equations in dimension $3$
I am trying to understand the dimensional reduction of Seiberg-Witten equations from dimension $4$ to $3$, more specifically my concern is about ellipticity of the new equations in dimension $3$ under ...
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0
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102
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Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields
Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection.
...
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3
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4
answers
3k
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References on principal G bundle and connections
I am trying to understand about principal G bundle given a Lie group $G$. For that, I started with the action of Lie groups on manifold $M$ and convinced myself that if the action is smooth, proper, ...
- 751
5
votes
1
answer
256
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Orientation bundle and its flat connection
Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any ...
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3
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705
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Modern treatment of Dirac monopoles and related topics
I know that the topic is classical and even "folklore", but many treatments make use of local coordinates and such treatments are rather messy. Could somewhere maybe provide some reference(s)...
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0
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0
answers
87
views
Confusion on a term related to connection and holonomy
This question is simply some of my confusions about a specific term.
Let $E\to X$ be a trivial complex vector bundle. When one says let $\nabla^E$ be a connection on $E\to X$ with trivial holonomy (...
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3
votes
1
answer
107
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Do we have an equivariant version of integrability theorem of flat connections?
I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1:
Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a ...
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0
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49
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Gauge-natural lifts of principal connections
Let $P=(P,\pi,M,G)$ be a principal fibre bundle and $\omega$ a principal connection on it. If $\lambda:G\times S\rightarrow S$ is a smooth left action of $G$ on a manifold $S$, the associated fibre ...
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4
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3k
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Can a manifold have a curvature-free connection that is not torsion-free?
Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-...
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3
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3k
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Is there a way to define a Lie derivative of a connection?
I've been reading a little bit about the definition of symmetries on General Relativity, and they are related with the concept of Killing vector, i.e., vectors along which the Lie derivative of the ...
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6
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0
answers
189
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What is a non-smooth connection?
Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\...
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5
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1
answer
820
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Coincide between Chern-connection and Levi-Civita connection
I am a beginner in complex geometry and I am going to show Levi-Civita connection $\nabla$ and the Chern connection $D$ are the same on the holomorphic tangent bundle $T^{1,0}M$ on Kahler manifold. By ...
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3
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Why torsion is only defined for linear connection on TM?
The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, ...
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6
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1
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463
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Holonomy bounded in terms of area and the curvature
I suppose the following result follows
from Ambrose-Singer theorem, but I cannot
find a reference, and the arguments I found
in the literature are usually weaker. The idea
is that holonomy over a null-...
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6
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1
answer
485
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What exactly is the relationship between an Ehresmann connection and splitting of the jet sequence?
An Ehresmann connection on a vector bundle $\pi : E \to X$ is a splitting of the sequence,
$$ 0 \to V \to TE \to \pi^* TX \to 0 $$
which respects the linear structure on $E$ (meaning the section is ...
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3
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1
answer
306
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Moduli space of flat connection over homology 3-sphere
I'm trying to understand the space of flat connections of the trivial $\mathrm{SU}(2)$-bundle over a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer ...
- 1,908
3
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0
answers
283
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Manifolds and Riemannian geometry with a bundle viewpoint
I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ...
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3
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0
answers
253
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Existence of connections in a vector bundle whose parallel transport preserves a function on a total space
Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space ...
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0
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56
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Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection
Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...
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7
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2
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Why does a principal G-bundle with a discrete structure group G have a unique flat connection?
I'm reading the Dijkgraaf–Witten paper Topological gauge theories and group cohomology (Comm. Math. Phys. 129 (1990) pp 393–429, doi:10.1007/BF02096988) and on page 395, 2nd paragraph they write
...
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2
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0
answers
71
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Covariant momenta associated to higher order Lagrangians
Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$).
Suppose that $L\in\Omega^m_{...
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3
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1
answer
297
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Torsion free (1,0)-connections on the holomorphic tangent bundle?
Let $M$ be a complex manifold. Consider a connection $\nabla$ on the holomorphic tangent bundle $T^{1,0}M$. The torsion of $\nabla$ is defined as the torsion of the induced connection $D$ on the real ...
- 26.3k
7
votes
1
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251
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What exactly is a Cartan radius vector (and its role in Poincaré gauge theories)
I am studying approaches to gravity where the Poincaré group is "gauged". The original motivation of this is to understand what is meant on the statement that "Teleparallel gravity is a gauge theory ...
CommunityBot
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0
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0
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171
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Koszul exterior connections
Let $(E,M)$ be a vector bundle over a riemannian manifold $M$ which is a module for the exterior forms of $M$. I define a Koszul exterior connection as an operator $\nabla$ such that:
$$
\nabla : E \...
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3
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0
answers
149
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Change of two normal coordinates based on two nearby points?
Let $M$ be a manifold and $L(M)$ be the tangent frame bundle on $M$. Let $\Gamma$ be a linear connection on $L(M)$ which induces a covariant derivative $\nabla$ on $TM$.
Let $p, q$ be two ...
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