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3 votes
1 answer
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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). \...
Ali Taghavi's user avatar
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 $M^n$, ...
ychemama's user avatar
  • 1,346
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 ...
Severin's user avatar
  • 105
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 ...
ychemama's user avatar
  • 1,346
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 ...
Praphulla Koushik's user avatar
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 $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
DLIN's user avatar
  • 1,915
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(...
Giovanni Moreno's user avatar
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 ...
AaronDefazio's user avatar
2 votes
2 answers
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 ...
M.G.'s user avatar
  • 7,127
8 votes
0 answers
286 views

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 ...
student's user avatar
  • 101
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}\...
Aureliano Skirzewski's user avatar
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 ...
Aureliano Skirzewski's user avatar
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 ...
Anthony Carapetis's user avatar
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 ...
truebaran's user avatar
  • 9,330
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, ...
Praphulla Koushik's user avatar
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 ...
user avatar
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 ...
Ali Taghavi's user avatar
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 ...
asierzm's user avatar
  • 51
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. ...
Bence Racskó's user avatar
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\...
Majid's user avatar
  • 227
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$\nabla$ ...
Zhaoting Wei's user avatar
  • 9,019
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 ...
Fallen Apart's user avatar
  • 1,615
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, ...
Bence Racskó's user avatar
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 ...
Asaf Shachar's user avatar
  • 6,741
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 ...
dorebell's user avatar
  • 3,058
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 ...
Miquel's user avatar
  • 11
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 ...
Raul's user avatar
  • 13
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<...
PtF's user avatar
  • 383
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,\...
C.F.G's user avatar
  • 4,195
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 ...
PtF's user avatar
  • 383
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 ...
C.F.G's user avatar
  • 4,195
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 (...
Ali Taghavi's user avatar
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:...
C.F.G's user avatar
  • 4,195
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 ...
Alex M.'s user avatar
  • 5,407
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 \...
Fly by Night's user avatar
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 \...
truebaran's user avatar
  • 9,330
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^...
David Roberts's user avatar
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 ...
Ho Man-Ho's user avatar
  • 1,173
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 ...
Tim Campion's user avatar
  • 63.9k
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/...
Brian Klatt's user avatar
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 ...
Saal Hardali's user avatar
  • 7,789
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 ...
Andrei Konyaev's user avatar
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 ...
Robin Goodfellow's user avatar
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 ^...
Yasha's user avatar
  • 491
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\...
Ettore Minguzzi's user avatar
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 ...
Dox's user avatar
  • 690
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 ...
ychemama's user avatar
  • 1,346
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: ...
Phillip's user avatar
  • 131
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 ...
Mauricio Tec's user avatar
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 $...
Bilateral's user avatar
  • 2,816