All Questions
Tagged with riemannian-geometry vector-bundles
17 questions with no upvoted or accepted answers
6
votes
0
answers
197
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Regarding a proof in the surgery theorem by Gromov and Lawson
I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is:
Gromov, ...
- 386
4
votes
0
answers
109
views
Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?
Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively ...
- 5,407
4
votes
0
answers
74
views
Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]
Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
- 3,051
3
votes
0
answers
157
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$N$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
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
201
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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
2
votes
0
answers
411
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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
$$TM \vert_{\...
- 21
2
votes
0
answers
192
views
Submanifold of Lie group whose tangent bundle is "almost" left-invariant
Let $G$ be a Lie group equipped with a left-invariant Riemannian metric, and let $M$ be a submanifold of $G$ containing the identity $e\in G$.
It is not difficult to show that, if the tangent bundle $...
- 965
2
votes
0
answers
35
views
Can one extend a Hermitian bundle from a compact manifold with boundary to its Riemannian double?
Let $M$ be a compact Riemannian manifold with boundary, and let $E \to M$ be a Hermitian vector bundle, endowed with a compatible connection. Let $\tilde M$ be a Riemannian double of $M$.
Does $E$ ...
- 5,407
2
votes
0
answers
68
views
Hypoellipticity or parabolic regularity for vector bundles
Let $E \to M$ be a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not necessarily compact). Let $H : \Gamma(E) \to \Gamma(E)$ be a differential operator with smooth coefficients ...
- 5,407
2
votes
0
answers
95
views
Vector bundle endomorphism diffeomorphism invariant?
Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
- 91
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
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
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
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 $p\...
- 11
1
vote
0
answers
180
views
Continuity of a convex function on a vector bundle
Consider the rank-${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
1
vote
0
answers
1k
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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