Quantum stochastic parallel transport on non-commutative vector bundles

1988 ◽  
pp. 20-36 ◽  
Author(s):  
David Applebaum
Keyword(s):  
Vector Bundles ◽  
Parallel Transport ◽  
Stochastic Parallel Transport

scholarly journals Generic Dynamical Properties of Connections on Vector Bundles

2021 ◽  
Author(s):  
Mihajlo Cekić ◽  
Thibault Lefeuvre
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Differential Operators ◽  
Parallel Transport ◽  
Conformal Killing ◽  
Generic Properties ◽  
Geometric Problems ◽  
Pseudo Differential Operators ◽  
Divergence Type ◽  
Open Question

Abstract Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla ^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted conformal Killing tensors (CKTs) are generically trivial when $\dim (M) \geq 3$, answering an open question of Guillarmou–Paternain–Salo–Uhlmann [ 14]. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations, which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla ^{\textrm{End}({\operatorname{{\mathcal{E}}}})}$ on the endomorphism bundle $\textrm{End}({\operatorname{{\mathcal{E}}}})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e., the geodesic flow is Anosov on the unit tangent bundle), the connections are generically opaque, namely that generically there are no non-trivial subbundles of $\mathcal{E}$ that are preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of Pollicott–Ruelle resonances in the Anosov case).

scholarly journals Parallel transport for vector bundles on $p$-adic varieties

2019 ◽  
Vol 29 (1) ◽  
pp. 1-52
Author(s):  
Christopher Deninger ◽  
Annette Werner
Keyword(s):  
Vector Bundles ◽  
Parallel Transport

CONNECTIONS, PARALLEL TRANSPORTS AND TRANSPORTS ALONG PATHS

2008 ◽  
Vol 05 (07) ◽  
pp. 1137-1162 ◽  
Author(s):  
BOZHIDAR Z. ILIEV
Keyword(s):  
Vector Bundles ◽  
Parallel Transport

The concepts "connection", "parallel transport" and "transport along paths" are axiomatically defined. In differentiable bundles, bijective mappings between the sets of connections, parallel transports and transports along paths satisfying some additional conditions are found; in particular, the equivalence between the concepts "connection" and "(axiomatically defined) parallel transport" is established. Some results, which are specific for topological and vector bundles, are presented.

scholarly journals Consistent curvature approximation on Riemannian shape spaces

2021 ◽  
Author(s):  
Alexander Effland ◽  
Behrend Heeren ◽  
Martin Rumpf ◽  
Benedikt Wirth
Keyword(s):  
Curvature Tensor ◽  
Vector Bundles ◽  
Parallel Transport ◽  
Time Discretization ◽  
Second Order ◽  
Shape Spaces ◽  
First Order ◽  
Covariant Derivatives ◽  
Low Dimensional ◽  
Curvature Approximation

Abstract We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end we extend the variational time discretization of geodesic calculus presented in Rumpf & Wirth (2015, Variational time discretization of geodesic calculus. IMA J. Numer. Anal., 35, 1011–1046), which just requires an approximation of the squared Riemannian distance that is typically easy to compute. First we obtain first-order discrete covariant derivatives via Schild’s ladder-type discretization of parallel transport. Second-order discrete covariant derivatives are then computed as nested first-order discrete covariant derivatives. These finally give rise to an approximation of the curvature tensor. First- and second-order consistency are proven for the approximations of the covariant derivative and the curvature tensor. The findings are experimentally validated on two-dimensional surfaces embedded in ${\mathbb{R}}^3$. Furthermore, as a proof of concept, the method is applied to a space of parametrized curves as well as to a space of shell surfaces, and discrete sectional curvature confusion matrices are computed on low-dimensional vector bundles.

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