Tensor

The user responsible for this page is Jost Eschenburg. No other user may edit this page at present.

This page has not been refereed. The information given here might be incomplete or provisional.

1 Definition

Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a smooth manifold and $E,E'$$E,E'$ vector bundles over $M$$M$. A tensor (field) is a section in the bundle $\textup{Hom}(E,E')$$\textup{Hom}(E,E')$ of bundle homomorphisms between $E$$E$ and $E'$$E'$. Alternatively, a tensor $T \in \Gamma\textup{Hom}(E,E')$$T \in \Gamma\textup{Hom}(E,E')$ can be viewed as a $C^\infty(M)$$C^\infty(M)$-linear map $T : \Gamma E \to \Gamma E'$$T : \Gamma E \to \Gamma E'$ which means

(1)$T(fs) = fT(s)$$T(fs) = fT(s)$

for any smooth function $f \in C^\infty(M)$$f \in C^\infty(M)$ and any section $s\in \Gamma E$$s\in \Gamma E$.

The bundle $E$$E$ may be itself a tensor product $E = E_1\otimes \dots\otimes E_k$$E = E_1\otimes \dots\otimes E_k$ of vector bundles $E_1,\dots,E_k$$E_1,\dots,E_k$. Then a tensor $T \in \textup{Hom}(E,E')$$T \in \textup{Hom}(E,E')$ may be viewed as a $C^\infty(M)$$C^\infty(M)$-multilinear map $T : \Gamma E_1 \times ...\times \Gamma E_k \to \Gamma E$$T : \Gamma E_1 \times ...\times \Gamma E_k \to \Gamma E$.