Tensor derivative

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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$$M$, both equipped with a covariant derivative $\nabla$$\nabla$. Then the vector bundle $\textup{Hom}(E,E')$$\textup{Hom}(E,E')$ of bundle homomorphisms (sometimes called tensors) inherits another covariant derivative $\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$$\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$ called the tensor derivative which is defined as follows:

(1)$(\nabla_XT)s = \nabla_X(Ts) - T(\nabla_Xs)$$(\nabla_XT)s = \nabla_X(Ts) - T(\nabla_Xs)$

for any $X \in \Gamma TM$$X \in \Gamma TM$ and $s\in \Gamma E$$s\in \Gamma E$. The definition is made such that the application of tensors (sections in $\textup{Hom}(E,E')$$\textup{Hom}(E,E')$) to sections in $E$$E$ satisfies the Leibniz product rule:

(2)$\nabla_X(Ts) = (\nabla_XT)s + T(\nabla_Xs)$$\nabla_X(Ts) = (\nabla_XT)s + T(\nabla_Xs)$

The corresponding curvature tensors of the bundles $E$$E$, $E'$$E'$ and $\textup{Hom}(E,E')$$\textup{Hom}(E,E')$ are related similarly:

(3)$(R(X,Y)T)s = R(X,Y)Ts - T(R(X,Y)s)$$(R(X,Y)T)s = R(X,Y)Ts - T(R(X,Y)s)$

for any $X,Y \in \Gamma TM$$X,Y \in \Gamma TM$ and $s\in \Gamma E$$s\in \Gamma E$.

A tensor $T \in \Gamma\Hom(E,E')$$T \in \Gamma\Hom(E,E')$ is called parallel if its tensor derivative vanishes, $\nabla_X T = 0$$\nabla_X T = 0$ for all $X \in \Gamma TM$$X \in \Gamma TM$.

2 Examples

• A (semi-)Riemannian metric $g \in \Gamma\Hom(S^2TM,\R)$$g \in \Gamma\Hom(S^2TM,\R)$ is parallel for its Levi-Civita connection.
• A Riemannian manifold has parallel curvature tensor $R \in \Gamma\Hom(\Lambda^2TM,\textup{End}(TM))$$R \in \Gamma\Hom(\Lambda^2TM,\textup{End}(TM))$ iff it is locally symmetric.
• A submanifold $M \subset \R^n$$M \subset \R^n$ has parallel second fundamental form $\alpha \in \textup{Hom}(S^2TM,NM)$$\alpha \in \textup{Hom}(S^2TM,NM)$ iff $M$$M$ is locally extrinsic symmetric.

For further information see [Husemoller1994] and [Kobayashi&Nomizu1963].

3 References

• [Husemoller1994] D. Husemoller, Fibre bundles, 3rd ed., Springer-Verlag, 1994. MR1249482 (94k:55001) Zbl 0794.55001
• [Kobayashi&Nomizu1963] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR1393940 (97c:53001a) Zbl 0508.53002