nLab Weitzenböck connection

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Context

Riemannian geometry

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 ℜ ⊣ ℑ infinitesimal ⊥ ⊥ infinitesimal ℑ ⊣ & 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)

On a (pseudo)Riemannian manifold the Weitzenböck connection is the affine connection on the tangent bundle which has vanishing curvature but possibly non-vanishing torsion.

This is in contrast to the Levi-Civita connection, for which it is the other way around.

Created on April 21, 2012 at 09:49:13. See the history of this page for a list of all contributions to it.

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