In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. Let π :...

vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields...

values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential...

Simpson. A Higgs bundle can be thought of as a "simplified version" of a flat holomorphic connection on a holomorphic vector bundle, where the derivative...

tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M {\displaystyle...

vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may...

a certain vector bundle over a base space S of a family of algebraic varieties V s {\displaystyle V_{s}} . The fibers of the vector bundle are the de...

information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under...

any fiber bundle associated to P {\displaystyle P} via the associated bundle construction. In particular, on any associated vector bundle the principal...

canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle...

to a vector bundle with flat connection as follows. The universal cover X ^ {\displaystyle {\hat {X}}} of X {\displaystyle X} is a principal bundle over...

{\displaystyle GL(n)} -bundle, the frame bundle. In particular, every smooth manifold has a canonical vector bundle, the tangent bundle. For a Lie group G...

In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold...

defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations...

the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry...

their current result is a formula that counts the Euler number of a flat vector bundle in terms of vertices of transversal open coverings. Notoriously, the...

says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it admits a flat unitary connection compatible with its...

flat through the points. Any vector space may be viewed as an affine space; this amounts to "forgetting" the special role played by the zero vector....

the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics...

{T} M} . Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each p in M, there are mutually inverse vector space isomorphisms...

In mathematics, a Lie algebroid is a vector bundle A → M {\displaystyle A\rightarrow M} together with a Lie bracket on its space of sections Γ ( A ) {\displaystyle...

covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra...

and (Nicolaescu 2002, 2003). If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the k-forms with...

the fiber is a vector space and the tensor bundle is a special kind of vector bundle. The vector bundle is a natural idea of "vector space depending...

canonical vector-valued 1-form on the frame bundle, the torsion Θ {\displaystyle \Theta } of the connection form ω {\displaystyle \omega } is the vector-valued...

system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of...

real formula for the bundle projection by noting that the fixed unit vector along the z axis, (0,0,1), rotates to another unit vector, ( 2 ( x z + w y )...

manifold M {\displaystyle M} is flat, there are simplifications which remove this subtlety. An associated vector bundle E → π M {\displaystyle E\xrightarrow...

τ, σ), where E is a vector bundle on XR, τ is a trivialization of E over (X \ x)R (i.e., an isomorphism with the trivial bundle O(X - x)R), and σ a trivialization...

holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the...