All Questions
49 questions
110
votes
6
answers
16k
views
When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion-free connection $\nabla_g$, the Levi-Civita connection, that is compatible with the metric.
I was wondering if one can ...
- 3,399
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
43
votes
5
answers
8k
views
A geometric interpretation of the Levi-Civita connection?
Let $M$ be a Riemannian manifold. There exists a unique torsion-free connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the Levi-...
- 23.5k
23
votes
3
answers
5k
views
Manifolds admitting flat connections
For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...
- 9,330
23
votes
4
answers
5k
views
Why is it important that partial derivatives commute?
I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...
- 995
20
votes
4
answers
3k
views
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-...
- 1,491
19
votes
6
answers
9k
views
Tensor contraction and Covariant Derivative
What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)?
In addition, I see that one of the requirements for a covariant ...
- 995
18
votes
3
answers
4k
views
Formal adjoint of the covariant derivative
Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...
- 5,824
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, ...
- 2,792
13
votes
2
answers
3k
views
Intuition for Levi-Civita connection?
Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
Question Is there some intuitively transparent constructive way to define it (or ...
- 24.9k
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 ...
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 ...
- 779
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 ...
- 4,195
11
votes
3
answers
886
views
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: ...
- 797
10
votes
1
answer
2k
views
Global description of the Levi-Civita connection
I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.
I'm not looking for a description of this ...
- 8,803
9
votes
2
answers
2k
views
Does every manifold have a flat connection?
Suppose I have a manifold and a vector bundle over it, but not a connection or a metric. Can I always find a connection on it that has a Riemann curvature tensor that is identically zero? If so, can I ...
- 1,491
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.
...
- 2,792
6
votes
1
answer
463
views
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-...
- 9,237
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
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 ...
- 9,330
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}\...
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 ...
- 690
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
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
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/...
- 787
4
votes
0
answers
166
views
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 ...
- 3,513
3
votes
6
answers
2k
views
The purpose of connections in differential geometry [closed]
I am currently reading through differential geometry as a mathematics graduate.
Can somebody give me a brief explainer on the purpose of connections?
I could also use explainers on differential forms. ...
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}$$
but the parallel ...
- 356
3
votes
1
answer
664
views
Definition of Levi-Civita connection map and a theorem about it?
Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem:
Theorem: If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and $K:...
- 4,195
3
votes
1
answer
191
views
Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$?
Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\...
- 356
3
votes
0
answers
102
views
Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields
Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection.
...
- 356
3
votes
0
answers
283
views
Manifolds and Riemannian geometry with a bundle viewpoint
I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ...
- 131
3
votes
0
answers
149
views
Change of two normal coordinates based on two nearby points?
Let $M$ be a manifold and $L(M)$ be the tangent frame bundle on $M$. Let $\Gamma$ be a linear connection on $L(M)$ which induces a covariant derivative $\nabla$ on $TM$.
Let $p, q$ be two ...
- 261
3
votes
0
answers
222
views
A question about a paper of Bismut and Lebeau
Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric.
Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
- 1,915
2
votes
1
answer
1k
views
When are geodesics straight lines?
Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients ...
- 213
2
votes
2
answers
277
views
Do "associative" connections exist / arise naturally in some context?
Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow.
Definition: Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative ...
- 7,127
2
votes
1
answer
232
views
If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?
This is a cross-post.
Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric.
Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...
- 6,741
2
votes
1
answer
182
views
Vector field along an immersion whose covariant derivative is the differential
Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one ...
- 2,247
2
votes
1
answer
485
views
A connection on $Hom( E,E)$ whose parallel transport is compatible to parallel transport of $E$
According to the answer of Sebastan and previous edit of Ben McKay I revise my post as follows:
Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$.
Is there a (...
- 356
2
votes
0
answers
411
views
Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates
Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle
$$TM \vert_{\...
- 21
2
votes
0
answers
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 ...
- 51
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\...
- 227
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: ...
- 131
1
vote
1
answer
197
views
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 \...
1
vote
1
answer
227
views
choices of connection in prequantization
In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...
user21574
1
vote
0
answers
153
views
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 ($x^a$ ...
- 250
1
vote
0
answers
632
views
Covariant Derivative of sections of a pullback bundle
Suppose that we have two smooth manifolds $M$ and $N$ and a smooth mapping $\phi : M \rightarrow N$. The Differential of that smooth mapping induces a bundle map $D\phi : TM \rightarrow TM$ between ...
- 5,327
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 ...
- 11
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,\...
- 4,195