nLab
tensor

Contents

This is about tensor quantities in the sense of multilinear algebra, differential geometry and physics, as in tensor calculus. For the different notion of a tensor in enriched category theory see under copower.


Context

Category theory

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

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Definition

          Generally, a tensor is an element of a tensor product.

          Traditionally this is considered in differential geometry for the following case:

          for XX a manifold, TXT X the tangent bundle, T *XT^* X the cotangent bundle, Γ(TX)\Gamma(T X), Γ(T *X)\Gamma(T^* X) their spaces of sections and C(X)C(X) the associative algebra of functions on XX, a rank-(p,q)(p,q) tensor or tensor field on XX is an element of the tensor product of modules over C(X)C(X)

          tΓ(TX) C(X) p C(X)Γ(T *X) C(X) q. t \in \Gamma(T X)^{\otimes_{C(X)}^p} \otimes_{C(X)} \Gamma(T^* X)^{\otimes^q_{C(X)}} \,.

          A rank (p,0)(p,0)-tensor is also called a covariant tensor and a rank (0,q)(0,q)-tensor a contravariant tensor.

          Examples

          General

          (…)

          In differential geometry

          References

          For instance section 2.4 of

          Last revised on April 20, 2016 at 05:00:18. See the history of this page for a list of all contributions to it.