All Questions
Tagged with dg.differential-geometry connections
201 questions
3
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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).
\...
6
votes
3
answers
1k
views
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 $M^n$, ...
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 ...
2
votes
1
answer
827
views
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 ...
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 ...
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
vote
1
answer
232
views
The bundle of symmetric affine connections as quotient of the second-order frame bundle
This post is not about finding an answer to a certain problem - because the answer already exists - but rather about finding the simplest possible answer.
The problem is: how to define the bundle $C(...
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 ...
2
votes
2
answers
277
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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 ...
8
votes
0
answers
286
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Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler
in a comment to this question When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?
Robert Bryant mentions that it is possible to construct a metric connection ...
4
votes
1
answer
303
views
Locally Riemannian Connection
Let $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ be a symmetric connection whose curvature is $$R^a{}_{bcd}=\partial_c\Gamma^a{}_{bd}-\partial_d\Gamma^a{}_{bc}+\Gamma^a{}_{ec}\Gamma^e{}_{bd}-\Gamma^a{}_{ed}\...
12
votes
1
answer
454
views
Riemannian vs Non-Riemannian curvature
If you neither know the metric nor the holonomy group, how do you recognize a curvature tensor is Riemannian?
I assume a curvature, by definition, satisfies Bianchi identities. I know it is ...
5
votes
0
answers
606
views
Is torsion of a connection always an obstruction to some kind of integrability? [closed]
Let $E$ be a vector bundle over a smooth manifold $M$ equipped with a linear connection $\nabla : \Gamma(E) \to \Omega^1(M;E).$ I say $(M,E,\nabla)$ is flat if it admits trivial local models; i.e. if ...
5
votes
2
answers
662
views
Most natural connection on Lie group: comparison of different pictures
Let $G$ be a Lie group (not necessarily compact). One can equip $G$ with the left invariant metric (or
right invariant but in general there is no biinvariant metric in the noncompact case). Once the ...
3
votes
4
answers
3k
views
References on principal G bundle and connections
I am trying to understand about principal G bundle given a Lie group $G$. For that, I started with the action of Lie groups on manifold $M$ and convinced myself that if the action is smooth, proper, ...
3
votes
2
answers
322
views
Affine connections as equivariant maps
An affine torsion-free connection on a smooth manifold $M$ may be thought of as a section of an affine bundle whose associated vector bundle is $S^2(T^*M)\otimes TM$. One would think that this affine ...
3
votes
1
answer
367
views
Is there such a connection on the punctured plane?
Is there a connection on $\mathbb{R}^2 \setminus \{0\}$ for which all operators of parallel transports are in the form $$\begin{pmatrix}a&-b\\b&a \end{pmatrix}$$
but the parallel ...
2
votes
0
answers
106
views
The dimension of the subspace of flat spin connections
I am interested in the the flat spin connections in a Riemann spacetime of dimension 4. They appear in the context of the frame formalism of metric gravity theories. I believe that they form a ...
7
votes
1
answer
2k
views
Reference for parallel transport around loop and its relation to curvature
It is a well known fact that the geometric meaning of a linear connection's curvature can be realized as the measure of a change in a fiber element as it is parallel transported along a closed loop.
...
2
votes
1
answer
80
views
Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$
I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\...
4
votes
0
answers
161
views
Do we have classical Riemann-Hilbert correspondence for infinite dimensional flat vector bundles?
Let $E$ be an $n$-dimensional vector bundle on a manifold $M$ and $\nabla: \Gamma(E)\to \Omega^1(M,E)$ be a flat connection on $E$. Classical Riemann-Hilbert correspondence tells us that ker$\nabla$ ...
8
votes
0
answers
480
views
Connections and curvature in commutative algebra
Since on any commutative algebra $R$ over ring $S$ we have module of Kahler differentials $(\Omega_{R/S},d)$ which extends to the algebraic de-Rham complex $(\Omega^\bullet,d),$ it is natural to ...
15
votes
2
answers
1k
views
When is a flow geodesic and how to construct the connection from it
Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following:
If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, ...
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 ...
7
votes
1
answer
2k
views
Can we define exterior derivatives using pushforwards and connections?
Let $\alpha$ be a differential form on a smooth manifold $M$. For simplicity, let's suppose that it is a $1$-form. Then we can think of $\alpha$ as a smooth map from $M$ to $T^* M$, the cotangent ...
1
vote
0
answers
1k
views
Splitting Short exact sequences of vector bundle with connection
Let $F\to M$ be a vector bundle and $E\subseteq F$ a subbundle. Suposse that $\nabla$ is a connection on $F$ s.t. preserves $E$, i.e. $\nabla_X(e)\in \Gamma E \quad \forall e\in \Gamma E, \ X\in\Gamma ...
1
vote
1
answer
244
views
Definition of Connection as G-invariant splitting of a sequence which is a pulled back sequence of bundles
I'm reading Atiyah-Bott's paper "The Yang-Mills equations over Riemann surfaces" and have a couple of questions on page 547. They define a connection $A$ as a $G$-invariant splitting of the exact ...
1
vote
0
answers
46
views
How to find $\beta^\prime_e(t)$ where $\beta_e(t)=\textrm{Hol}^\sigma_{\gamma_{1, t}}(e)$?
Let $p:E\longrightarrow B$ be a surjective submersion and $\sigma: p^*(TB)\longrightarrow TE$ a complete connection. Given a path $\gamma: [a, b]\longrightarrow B$ and $s, t\in [a, b]$ such that $s<...
0
votes
1
answer
134
views
On generalized Tanaka connection
Many authors used the Tanaka connection in their papers such as
[1]
to define new Tanaka connection so-called Generalized Tanaka connection $^*\nabla$ on a contact Riemannian manifold $(M,\eta,\xi,\...
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 ...
11
votes
1
answer
7k
views
Geometric interpretation of horizontal and vertical lift of vector field
In many References such as D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds chapter 9, and Differential Geometric Structures
By Walter A. Poor Page 54; the horizontal and vertical ...
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 (...
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:...
1
vote
1
answer
210
views
Hermitic connections on complex line bundles with imaginary curvature form
It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...
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 \...
9
votes
1
answer
693
views
Generalized Dirac operators
So far I met three definitions of the so called generalized Dirac operator(or Dirac type operators. Everything takes place over Riemannian manifols $M$ and we have smooth hermitian vector bundle $S \...
1
vote
1
answer
229
views
Is a non-flat hermitian connection determined uniquely by its holonomy and curvature?
How do I prove that gauge-equivalence classes of $U(1)$ connections on a line bundle $L\to M$ are determined uniquely by pairs $(\alpha,F)$, where
$$\alpha\in\text{Hom}(\pi_1(M),U(1)),~~~~F\in \Omega^...
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 ...
4
votes
2
answers
505
views
Holonomy of a Ricci-flat affine connection
There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently ...
4
votes
1
answer
382
views
Parallel Transport on Hypersurface Spinor Bundle
So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link:
https://projecteuclid.org/...
6
votes
1
answer
2k
views
Transferring connection information to associated bundles and back
This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try.
At the risk of repeating well known stuff I tried ...
4
votes
1
answer
300
views
Connection, compatible with type (1, 1) tensor field
I met with the following problem. Consider real manifold $M^{2n}$ with operator field $R$ (that is the tensor field of type $(1,1)$). We are to find a symmetric connection $\Gamma^k_{ij}$ such that ...
12
votes
1
answer
937
views
Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?
I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...
3
votes
0
answers
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 ^...
3
votes
1
answer
186
views
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\...
4
votes
1
answer
412
views
Symmetries of non-Riemannian curvature tensor
The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...
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
vote
3
answers
572
views
Special connection of vector bundle over real manifold
Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...
4
votes
2
answers
419
views
What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?
Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...
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 $...