nLab vertical differential form

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

In differential geometry

Let π:PX\pi : P \to X be a bundle in the category Diff of smooth manifolds.

The dg-algebra Ω vert (P)\Omega^\bullet_{vert}(P) of vertical differential forms on PP is the quotient of the de Rham complex dg-algebra Ω (P)\Omega^\bullet(P) of all forms on PP, by the dg-ideal of horizontal differential forms, hence of all those forms that vanish when any one vector in their arguments is a vertical vector field in that it is in the kernel of the differential dπ:TPTXd \pi : T P \to T X.

For a trivial bundle P=X×FP = X \times F the underlying complex of Ω vert (P)\Omega^\bullet_{vert}(P) is C (X×F) Γ(T *F)\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F).

References

  • Philippe Bonnet, Alexandru Dimca, Relative differential forms and complex polynomials, Bulletin des Sciences Mathématiques 124 Issue 7 (2000) pp 557-571, doi:10.1016/s0007-4497(00)01055-1, (author pdf)

Last revised on September 19, 2023 at 05:17:14. See the history of this page for a list of all contributions to it.