All Questions
Tagged with dg.differential-geometry connections
201 questions
43
votes
5
answers
9k
views
What is the Levi-Civita connection trying to describe?
I have seen similar questions, but none of the answers relate to my difficulty, which I will now proceed to convey.
Let $(M,g)$ be a Riemannian manifolds. The Levi-Civita connection is the unique ...
- 2,071
1
vote
0
answers
90
views
Curvature of a superconnection
Let $E\rightarrow X$ be a $\mathbb{Z}_2$-vector bundle (or superbundle for connoisseurs) and consider the superconnection
$$A=\nabla + B$$
where $\nabla$ is a connection on $E$ and $B\in\Gamma(End(E))^...
- 789
9
votes
0
answers
231
views
Is there a contact instanton connection on the tangent bundle of the 5-sphere?
A well-known example of a contact manifold is $S^5$, arising from it being a circle bundle over $\mathbb{CP^2}$. This is somewhat related to the reduction of the structure group from $SO(5)$ to $SU(2)$...
- 35.5k
6
votes
4
answers
1k
views
Connections in the setting of algebraic geometry
My level is at the beginning of a second year master. I'm interested in the project of translating some features of differential geometry to algebraic geometry. I'd like to know if there is an ...
1
vote
1
answer
289
views
What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bundle?
$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $\At(P)$ of a ...
- 1,215
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
1
vote
0
answers
195
views
Can logarithmic connection on holomorphic vector bundle induce logarithmic connection on dual bundle?
Let $(X,\omega)$ be a compact K"ahler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting ...
- 295
6
votes
0
answers
297
views
Examples of connection preserving maps in differential geometry
In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other ...
- 1,253
5
votes
0
answers
311
views
Hopf fibration extended to bundle over $\mathbb{C}^2$
Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...
2
votes
1
answer
929
views
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 ...
- 6,023
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
6
votes
1
answer
352
views
What is the definition of homotopy flat connections?
What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra
- 3,880
6
votes
0
answers
156
views
Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$
An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below:
Characteristic classes are certain cohomology classes associated
...
- 6,023
3
votes
0
answers
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
2
votes
1
answer
518
views
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
votes
0
answers
47
views
Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates
In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented.
Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
- 1,155
5
votes
0
answers
466
views
A struggle with jets and Grothendieck vs Ehresmann connections
Let $X\to Y$ be a $C^\infty$ submersion. Consider the following two sheaves.
The sheaf on $Y$ comprised of jets of sections of $X\to Y$.
The sheaf on $X$ given by the quotient of $\Delta_{X/Y}^{-1}C^\...
- 10.5k
6
votes
0
answers
375
views
Is there an analog of the Levi–Civita connection for schemes?
Is there an analog of the Levi–Civita connection for schemes?
There exists algebraic de Rham theory, $n$-forms on vector bundles (algebraically), and familiar constructions from differential geometry....
- 327
4
votes
1
answer
132
views
Problem arising in metrizability of connections: Simultaneously skewsymmetrizing matrices
Fact: Let $U$ and $V$ be two $ n \times n$ matrices with determinant $ 1.$ Assume that $S_1,S_2,....S_m$ are linearly independent $n \times n$ matrices such that $U^{-1}S_iU$ and $V^{-1}S_iV$ are ...
- 463
7
votes
2
answers
2k
views
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
...
- 569
3
votes
1
answer
2k
views
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
1
vote
0
answers
409
views
Introducing connection on principal bundle as lifting of vector field and paths
Let $\pi:P\rightarrow M$ is a principal $G$ bundle. I want to introduce the notion of connection as a way to uniquely lift the structures on $M$ to structures on $P$, namely vector fields and paths.
...
- 6,023
1
vote
1
answer
1k
views
Existence of horizontal lifts in $G$-bundles
I wanted to show that for any smooth principal $G$-bundle $E\xrightarrow\pi B$ any smooth curve $\gamma\colon I\to B$ has a unique horizontal lift from a fixed starting point $u_0\in\pi^{-1}\left(\...
- 438
5
votes
1
answer
560
views
Holonomy map on a connected manifold determines the connection and the bundle
I am reading Parallel transport on principal bundles over stacks. I quote from their paper :
Let $G$ be a Lie group and $M$ a $C^{\infty}$ manifold. Recall that a
choice of a connection $1$-form ...
- 6,023
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
1
vote
0
answers
337
views
Definition of an equivariant connection and equivariant curvature
Can anyone give me a reference which precisely stated the definition of an equivariant connection and equivariant curvature?
Precisely, If G be a compact lie group acting linearly on a smooth ...
0
votes
1
answer
176
views
What is the group of symmetries of $\mathbb{R^n}$ with the flat projective structure?
Consider $X = (\mathbb{R^n},c)$, where $c$ is the equivalence class of all torsion free affine connections having straight lines as unparameterized geodesics. What is the group of symmetries of $X$? ...
- 5,215
4
votes
0
answers
118
views
Flatness equivalence
Let $\pi:E\rightarrow M$ be a complex vector bundle and $H$ a hermitian metric over it. If $D$ is a connection over $E$, using the metric $H$, we can decompose it as:
$$
D=D_H+\phi
$$
Where $D_H$ is ...
- 311
4
votes
1
answer
1k
views
Confusion about complex differential forms
I follow Kobayashi "Differential Geometry of Complex Vector Bundles", pages 11-12, prop. 4.9. Given a rank-$r$ Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $M$, there exists a ...
- 143
7
votes
1
answer
296
views
ASD connection for Line bundle over $4$-manifold
Let $(M,g)$ be an oriented closed Riemannian $4$ manifold.
Let $L\to M$ be a complex line bundle.
Q Under what condition, we can find an ASD connection of $L$, i.e. a connection $A$ such that $F^+...
- 1,915
4
votes
2
answers
307
views
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive ...
- 311
3
votes
2
answers
653
views
When is the action of the gauge group on the space of connections free?
Let $G$ be a compact Lie group. Let $\mathcal{A}$ be the space of connections on the principal trivial $G$-bundle $G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of ...
- 21.8k
7
votes
1
answer
251
views
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 ...
- 2,792
2
votes
0
answers
76
views
Integrability of connections with partially vanishing curvature
Let $E \rightarrow B$ be a vector bundle with a connection $\nabla$ and a local frame $(e_1, \dots, e_n)$. For any section $V = V^a e_a$ the connection can locally be written as
$$
\nabla V = \left(d ...
- 105
7
votes
3
answers
2k
views
Atiyah Sequence and Connections on a Principal Bundle
Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.
I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence
$$ 0\rightarrow \text{...
- 6,023
4
votes
1
answer
259
views
Metrics with prescribed Levi-Civita connection
My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the ...
- 1,329
5
votes
0
answers
115
views
Flattening a connection on a Kähler manifold
Say $M$ is a closed Kähler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ Kähler gives several distinguished classes of closed one-...
- 7,952
4
votes
0
answers
400
views
Chern-Weil theory and Weil homomorphism of principal bundle
In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to ...
- 6,023
5
votes
1
answer
253
views
Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$?
This is a cross-post.
Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\...
- 6,741
1
vote
1
answer
266
views
Flat covariant derivative
Is it true that for any flat and torsion-free connection $\nabla : \mathfrak{X} (M) \times \mathfrak{X} (M) \rightarrow \mathfrak{X} (M) $ there exist a local systems of coordinates such that the ...
- 11
5
votes
1
answer
208
views
Inducing linear connections via functors
Let $M$ be a smooth manifold and let $\pi:E\rightarrow M$ be a real vector bundle over it. Let $\nabla$ be a linear (Koszul) connection on $E$ (here in this question I am using covariant derivatives, ...
- 2,792
1
vote
2
answers
458
views
Is there a notion of a connection for which the horizontal lift of a curve depends on its orientation?
Given a fiber bundle $\pi:E\to M$, a curve $\gamma:[0,1]\to M$, and a point $p \in \pi^{-1}(\gamma(0))$, a connection on the bundle allows us to uniquely lift $\gamma$ to a horizontal curve in E ...
- 245
2
votes
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
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
1
vote
1
answer
224
views
determinant of curvature (notation issue)
This is when studying about Chern classes from Kobayashi and Nomizu.
Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$.
Let $p:P\rightarrow ...
- 6,023
3
votes
0
answers
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
votes
4
answers
3k
views
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 $1$ ...
- 6,023
3
votes
0
answers
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
answers
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
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