Everything about Connection (vector bundle)
Defines a notion of parallel transport on a bundle
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Connection (vector bundle)A 5-minute orientation
- Skim a recent paper: Smooth one-dimensional topological field theories are vector bundles with connectionInfluential work since 2022
- Pick up a classic book: A theory of characteristic currents associated with a singular connectionThe most-reprinted book on the subject
- Read the landmark paper: Generalised connections over a vector bundle mapCited 15 times
- Go sideways: Metric connectionA closely related topic
- Go sideways: Covariant derivativeA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Generalised connections over a vector bundle map
On the inverse problem of determining a connection on a vector bundle
Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree
A criterion for the existence of a flat connection on a parabolic vector bundle
Geometric phase for mixed states: Formulation based on a connection on a vector bundle over a Grassmann manifold
Connection of topological vector bundles
A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions
Non-linear connection in vector bundles
🆕 Influential since 2022
Smooth one-dimensional topological field theories are vector bundles with connection
Classification of Vector Fields and Soliton Structures on a Tangent Bundle with a Ricci Quarter-Symmetric Metric Connection
Invariants of stably trivial vector bundles with connection
On the semi-finite vector bundles with connection over Kähler manifolds
📚 Books worth reading
- 📕A theory of characteristic currents associated with a singular connectionF. Reese Harvey · 1993
- 📕Grassmannians, moduli spaces, and vector bundlesClay Mathematics Institute Workshop on Moduli Spaces of Vector Bundles, with a View toward Coherent Sheaves (2006 Cambridge, Mass.) · 2011
General theory of lie groupoids and lie algebroidsK. Mackenzie · 2005- 📕Modern GeometryVicente Munoz, Ivan Smith · 2018
- 📕Analysis, complex geometry, and mathematical physicsDuong H. Phong, Paul M. N. Feehan · 2015
- 📕Nine Mathematical Challenges : An ElucidationA. S. Kechris, Nikolai G. Makarov · 2021
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Metric connection In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of…
- Covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative…
- Laplace operators in differential geometry In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides…
- Affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector…
- Connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving…
- Parallel transport In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If…
- Ehresmann connection In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of…
- Tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space…