All Questions
Tagged with dg.differential-geometry connections
201 questions
3
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1
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367
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Is there such a connection on the punctured plane?
Is there a connection on $\mathbb{R}^2 \setminus \{0\}$ for which all operators of parallel transports are in the form $$\begin{pmatrix}a&-b\\b&a \end{pmatrix}$$
but the parallel ...
- 356
3
votes
1
answer
456
views
Left invariant connections on a Lie group
The exponential map associated with the (-) - connection on a Lie group is generally not surjective. This is because, for this connection, the one-parameter subgroups and geodesics coincide. If we ...
- 1,378
3
votes
2
answers
375
views
holonomy of connection on gerbes
I am reading this notes of Hitchin to understand about gerbes. He defines gerbe by giving a collection of $2$ cocycles $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ with ...
- 6,023
3
votes
2
answers
322
views
Affine connections as equivariant maps
An affine torsion-free connection on a smooth manifold $M$ may be thought of as a section of an affine bundle whose associated vector bundle is $S^2(T^*M)\otimes TM$. One would think that this affine ...
user113646
3
votes
1
answer
507
views
Local coordinates of one form on a principal bundle
I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.
Let's say ...
- 247
3
votes
1
answer
2k
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Flat connections, curvature and holonomy
Let $A$ be a flat connection on a principal $G$-bundle $G\hookrightarrow P\to M$.
Consider an homotopically trivial loop $\gamma \subset M$. For simplicity, suppose $\gamma = \partial D$ is the ...
- 2,533
3
votes
1
answer
664
views
Definition of Levi-Civita connection map and a theorem about it?
Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem:
Theorem: If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and $K:...
- 4,195
3
votes
1
answer
955
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Holonomy Groups and the Hopf Fibration
I am trying to understand holonomy groups at the moment and am focusing on the example of the Hopf fibration $SU_2 \to S^2$. Since $S^2$ is path connected we can talk about the holonomy group of a ...
- 3,399
3
votes
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 ...
- 181
3
votes
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 ...
- 1,457
3
votes
1
answer
191
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Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$?
Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\...
- 356
3
votes
1
answer
102
views
Comparing holonomies along different connections?
Let $p:E\longrightarrow B$ be a smooth surjective submersion and $\sigma, \sigma^\prime: p^*(TB)\longrightarrow TE$ be two complete connections. Given a path $\gamma:I\longrightarrow B$ we can ...
- 383
3
votes
1
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186
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Are non-linear connections with linear holonomy, linear?
Let $\pi\colon TM\to M$ be the tangent bundle of a differentiable manifold, let $E=TM\backslash 0$ be the slit tangent bundle, and let $V_eE$ be the kernel of $\pi_*$ at $e\in E$. The set $VE=\cup_{e\...
- 1,371
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 ...
- 9,019
3
votes
1
answer
567
views
Projectively flat connection
Let $E \to B$ be a Hermitian vector bundle. If $E$ has a projectively flat connection, then its total Chern character has the form $\mbox{ch}(E) = \mbox{rank} \cdot \exp(\mbox{slope})$. Is the ...
- 363
3
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0
answers
89
views
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 ...
- 661
3
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0
answers
102
views
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.
...
- 356
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 ...
- 131
3
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0
answers
149
views
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 ...
- 261
3
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0
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522
views
Writing a Taylor series with covariant derivatives (connections)?
A connection of a vector bundle $E$ on a manifold $M$ is a map $d_E: \Omega^0(E) \to \Omega^1(E)$ that extends uniquely to a map $d_E: \Omega^p(E) \to \Omega^{p+1}(E)$ while satisfying
$$
d_E(\omega \...
- 1,755
3
votes
0
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445
views
Pullback connection and diffeomorphism of the base
Let $p \colon E \to B$ be a vector bundle, $\nabla^E \colon E \to E \otimes \Omega^1_B$ a connection on $E$, and $\phi \colon B \xrightarrow{\sim} B$ a diffeomorphism. Further, let there be a natural (...
- 949
3
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0
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155
views
Maximally symmetric affine manifold
As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $n$-dimensional manifold with $\frac{n(n+1)}{2}$ Killing vectors. A ...
- 690
3
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0
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476
views
Applications of Ambrose-Singer theorem on holonomy
I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. I want to conclude with the statement of Ambrose-Singer ...
3
votes
0
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253
views
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 ...
3
votes
0
answers
436
views
Existence of sections of a fibre bundle which are covariantly constant along certain directions
Given a vector bundle $\pi\colon E \rightarrow B$ equipped with a connection $\nabla$, it is well known that a basis of flat sections $s_i$ ($i=1,\dots,\text{rank}(E)$) (i.e. $\nabla_X s_i = 0$ for ...
- 105
3
votes
0
answers
222
views
A question about a paper of Bismut and Lebeau
Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric.
Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
- 1,915
3
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0
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147
views
Growth of norm of curvature under direct sum or existence of universal connection
For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by:
$$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ^...
- 491
3
votes
0
answers
483
views
Connection and reduction of the structure group
I apologize for this question which is not really research-level, but I don't get any answer on mathStackExchange, and I asked a professor in my university... which couldn't find the solution.
I am ...
- 1,346
3
votes
0
answers
238
views
Parallel Ricci condition - Status report and bibliography
First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...
- 690
2
votes
1
answer
1k
views
When are geodesics straight lines?
Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients ...
- 213
2
votes
2
answers
869
views
A question about flat connection
Let $E\to X$ be a complex flat vector bundle, and say $\nabla_0$ and $\nabla_1$ are two flat connections on it. Let $p:X\times[0, 1]\to X$ denote the projection onto the first factor. Is there a way ...
- 1,173
2
votes
2
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277
views
Do "associative" connections exist / arise naturally in some context?
Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow.
Definition: Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative ...
- 7,127
2
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1
answer
232
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If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?
This is a cross-post.
Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric.
Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...
- 6,741
2
votes
1
answer
454
views
Advantages of Atiyah sequence version of connections on a principal bundle
I am reading Lie Groupoids and Lie Algebroids in Differential Geometry
by Kirill Mackenzie.
In appendix (page $291$), before discussing about Atiyah sequence associated to a Principal bundle, the ...
- 6,023
2
votes
1
answer
670
views
Metric, torsion free connections on principal bundles
I hope this is not too elementary, but I have asked this question at the math.stack site, but I have obtained no answers.
Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $...
- 2,816
2
votes
1
answer
182
views
Vector field along an immersion whose covariant derivative is the differential
Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one ...
- 2,247
2
votes
1
answer
271
views
Canonical connection on $\mathcal{A}\times X$
Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
- 789
2
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1
answer
518
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Katz's paper on $p$-curvature – help with proof understanding
I am studying N. Katz's paper "Nilpotent connections and the monodromy theorem: applications of a result of Turrittin" where I found a fairly good account on $p$-curvatures.
I don't understand the ...
- 339
2
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1
answer
827
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Connection 1-form of the frame bundle associated to a vector bundle with a connection
Let $\lambda = (P,\pi,M;G)$ be a smooth principal $G$-bundle (projection $\pi : P \to M)$, $V$ a finite dimensional vector space, and $\rho : G \to GL(V)$ a smooth representation of $G$ in $V$.
We ...
- 1,346
2
votes
1
answer
279
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Immersion with respect to a connection?
In the paper hep-th/9712042v2, p. 20, the following setup is given:
A complex manifold M and an n+1-dimensional vector bundle V on it. V has an underlying real bundle $V_{\mathbb{R}}$ with a flat ...
- 123
2
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1
answer
415
views
The exterior derivative of a certain differential form on the space of connections of a surface
Let $Y$ be a closed oriented $2$-dimensional manifold, $G$ a Lie group and $Q \to Y$ a principal $G$-bundle with a given section $q.$ Denote by $\mathcal{A}_Q$ the space of connections on $Q,$ and by $...
user44591
2
votes
2
answers
661
views
Does "symmetry" of a pullback connection should be obvious?
$\newcommand{\M}{M}$
$\newcommand{\N}{N}$
$\newcommand{\TM}{TM}$
$\newcommand{\TN}{TN}$
$\newcommand{\TstarM}{T^*M}$
$\newcommand{\Ga}{\Gamma}$
Let $\M,\N$ be smooth manifolds, $\phi:\M \to \N$ be a ...
- 6,741
2
votes
1
answer
485
views
A connection on $Hom( E,E)$ whose parallel transport is compatible to parallel transport of $E$
According to the answer of Sebastan and previous edit of Ben McKay I revise my post as follows:
Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$.
Is there a (...
- 356
2
votes
1
answer
398
views
Flatness as an integrability condition without invoking bundles
Let $\pi:E\rightarrow M$ be a vector bundle, and let $U\subseteq M$ be a trivialization domain for $E$. Assume a linear connection is given on $E$ with local connection form(s) $\omega=(\omega^a_{\ b})...
- 2,792
2
votes
1
answer
143
views
Proof about affine connections
I'm reading Nomizu & Sasaki's "Affine Differential Geometry: Geometry of Affine Immersions" and I'm having some trouble with Proposition 1.4.
I have an immersed surface in $M \hookrightarrow \...
- 310
2
votes
1
answer
103
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Normalizing the value of a principal connection at a point
Let $\nabla$ be a symmetric, linear connection on a smooth manifold $X$.
If $p \in X$ is any point, on a normal chart for $\nabla$ around $p$ it holds:
$$ \Gamma_{ij}^k (p) = 0 \ , $$ where $\Gamma_{...
- 1,064
2
votes
1
answer
171
views
Construction of a classifying map from a connection 1-form
From a connection 1-form on $M$, I can construct a parallel transport from which in turn I can construct a classifying map $M \to BG$.
Is there a construction of such a classifying map directly from a ...
- 3,880
2
votes
0
answers
53
views
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 (...
- 243
2
votes
0
answers
411
views
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_{\...
- 21
2
votes
0
answers
169
views
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 ...
- 954