All Questions
30 questions
5
votes
1
answer
464
views
Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
- 1,171
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
3
votes
0
answers
157
views
$N$th-order approximation of point stabilizing diffeomorphisms by $N$th-order jet group?
NOTE: migrated from math SE.
I was wondering if ever higher jet groups of frames on a (possibly pseudo) Riemannian manifold $M$ approximate the point stabilizing subgroup of diffeomorphisms on $M$ as ...
- 250
4
votes
1
answer
334
views
Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief ...
- 4,195
3
votes
1
answer
470
views
Is the Moebius strip Riemannian homogeneous?
Let $ M $ be the Moebius band. In other words, the total space of the nontrivial line bundle over the circle. Can we equip $ M $ with a metric such the the isometry group acts transitively?
My ...
- 4,081
3
votes
1
answer
292
views
Identification of tangent spaces by parallel transport along geodesics [closed]
Given a geodesically complete manifold M, can we define a global identification of tangent spaces by starting from a base point, and parallel transporting along smooth geodesics? For this to be ...
- 437
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
2
votes
1
answer
138
views
noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous
Is it true that a manifold $ E $ admits a metric with respect to which the isometry group is transitive ($ E $ is Riemannian homogeneous) if and only if $ E $ is the total space of a $ K $ equivariant ...
- 4,081
2
votes
0
answers
192
views
Submanifold of Lie group whose tangent bundle is "almost" left-invariant
Let $G$ be a Lie group equipped with a left-invariant Riemannian metric, and let $M$ be a submanifold of $G$ containing the identity $e\in G$.
It is not difficult to show that, if the tangent bundle $...
- 985
0
votes
1
answer
108
views
Intersection Grassmanian planes
I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
- 1,043
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
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
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
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
2
votes
0
answers
35
views
Can one extend a Hermitian bundle from a compact manifold with boundary to its Riemannian double?
Let $M$ be a compact Riemannian manifold with boundary, and let $E \to M$ be a Hermitian vector bundle, endowed with a compatible connection. Let $\tilde M$ be a Riemannian double of $M$.
Does $E$ ...
- 5,407
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
2
votes
0
answers
197
views
Existence of a certain kind of compact spin manifold with boundary
For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
- 121
4
votes
2
answers
514
views
Is the kernel of the coderivative infinite-dimensional?
$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$
Let $M,N$ be smooth $n$-dimensional Riemannian manifolds (...
- 6,741
2
votes
0
answers
95
views
Vector bundle endomorphism diffeomorphism invariant?
Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
- 91
3
votes
0
answers
201
views
The heat kernel in Hermitian bundles over Riemannian manifolds
In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....
- 5,407
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
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
1
vote
0
answers
225
views
Extending fibre metrics of submanifolds to Riemannian metrics
Let $M$ be a smooth manifold and $S\subseteq M$ a properly embedded smooth submanifold. Suppose that we have a fibre metric on $TM|_S$, i.e. a positive definite real inner-product on $T_pM$ for all $p\...
- 11
2
votes
2
answers
124
views
Invertible (isometric) sections of certain hom bundles over sphere
Assume that we have a vector bundle $E$ over $S^n$.
Is there a continuous family of invertible linear maps $T_x:E_x \to E_{-x}$?
Here continuity has the obvious meaning as soon as ...
- 356
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
1
vote
1
answer
317
views
Parallel transport in Riemannian manifold induces bounded mapping of vector bundles
Let $X$ and $Y$ be closed Riemannian manifolds and $f,g\colon X\to Y$ two $C^1$-mappings.
Assume that for every $x\in X$ the points $f(x)$ and $g(x)$ can be joined by a unique shortest geodesic of $...
- 13
1
vote
0
answers
180
views
Continuity of a convex function on a vector bundle
Consider the rank-${n \choose m}$ vector bundle $\pi\colon E:=\bigwedge^m(TN)\to N$ over a smooth Finsler manifold $N$ and equip each fibre $E_q := \pi^{-1}(q)$ with a norm that depends smoothly on $q\...
- 193
2
votes
0
answers
250
views
Equivariant vector bundle structure on the tangent bundle of compact Riemann surfaces with non trivial action on the base space,
Let $M_{g}$ be the compact Riemann surface with $g\geq 2$.
Is there an infinit group $G$ with an equivariant action on the pair $(TM_{g}, M_{g})$ such that the action on the fibers preserves the inner ...
- 356
0
votes
2
answers
374
views
On the definition of convergence of a sequence of sections of a bundle
Convergence of a sequence of sections of a bundle is defined as follows:
Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on $TM$...
- 1,303
5
votes
1
answer
1k
views
Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
- 8,803