Tensor derivative
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$\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$ | $\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$ | ||
called the ''tensor derivative'' which is defined as follows: | called the ''tensor derivative'' which is defined as follows: | ||
− | + | \begin{equation} (\nabla_XT)s = \nabla_X(Ts) - T(\nabla_Xs) \end{equation} | |
for any $X \in \Gamma TM$ and $s\in \Gamma E$. | for any $X \in \Gamma TM$ and $s\in \Gamma E$. | ||
The definition is made such that the application of | The definition is made such that the application of | ||
tensors (sections in $\textup{Hom}(E,E')$) to sections in $E$ satisfies the Leibniz product rule: | tensors (sections in $\textup{Hom}(E,E')$) to sections in $E$ satisfies the Leibniz product rule: | ||
− | + | \begin{equation} \nabla_X(Ts) = (\nabla_XT)s + T(\nabla_Xs) \end{equation} | |
The corresponding [[Curvature tensor and second derivative|curvature tensors]] of the bundles $E$, $E'$ and $\textup{Hom}(E,E')$ are related similarly: | The corresponding [[Curvature tensor and second derivative|curvature tensors]] of the bundles $E$, $E'$ and $\textup{Hom}(E,E')$ are related similarly: | ||
− | + | \begin{equation} (R(X,Y)T)s = R(X,Y)Ts - T(R(X,Y)s) \end{equation} | |
for any $X,Y \in \Gamma TM$ and $s\in \Gamma E$. | for any $X,Y \in \Gamma TM$ and $s\in \Gamma E$. | ||
+ | |||
+ | A tensor $T \in \Gamma\Hom(E,E')$ is called ''parallel'' if its tensor derivative vanishes, $\nabla_X T = 0$ | ||
+ | for all $X \in \Gamma TM$. | ||
</wikitex> | </wikitex> | ||
+ | == Examples== | ||
+ | <wikitex>; | ||
+ | * A (semi-)Riemannian metric $g \in \Gamma\Hom(S^2TM,\R)$ is parallel for its [[Levi-Civita connection|Levi-Civita connection]]. | ||
+ | * A Riemannian manifold has parallel curvature tensor $R \in \Gamma\Hom(\Lambda^2TM,\textup{End}(TM))$ iff it is locally symmetric. | ||
+ | * A submanifold $M \subset \R^n$ has parallel second fundamental form $\alpha \in \textup{Hom}(S^2TM,NM)$ iff $M$ is locally extrinsic symmetric. | ||
+ | For further information see \cite{Husemoller1994} and \cite{Kobayashi&Nomizu1963}. | ||
+ | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Definitions]] | [[Category:Definitions]] | ||
+ | [[Category:Connections and curvature]] |
Latest revision as of 10:37, 23 May 2013
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1 Definition
Let be a smooth manifold and vector bundles over , both equipped with a covariant derivative . Then the vector bundle of bundle homomorphisms (sometimes called tensors) inherits another covariant derivative called the tensor derivative which is defined as follows:
for any and . The definition is made such that the application of tensors (sections in ) to sections in satisfies the Leibniz product rule:
The corresponding curvature tensors of the bundles , and are related similarly:
for any and .
A tensor is called parallel if its tensor derivative vanishes, for all .
2 Examples
- A (semi-)Riemannian metric is parallel for its Levi-Civita connection.
- A Riemannian manifold has parallel curvature tensor iff it is locally symmetric.
- A submanifold has parallel second fundamental form iff 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