Covariant derivative

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1 Definition

Let $E \to M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}E \to M$ be a vector bundle over a smooth manifold $M$$M$ and $\Gamma E$$\Gamma E$ the space of smooth sections. A covariant derivative on $E$$E$ is a bilinear map $\nabla : \Gamma TM \times \Gamma E \to \Gamma E$$\nabla : \Gamma TM \times \Gamma E \to \Gamma E$, $(X,s)\mapsto\nabla_Xs$$(X,s)\mapsto\nabla_Xs$, which is a tensor (linear over $C^\infty(M)$$C^\infty(M)$) in the first argument and a derivation in the second argument:

(1)$\begin{matrix}\nabla_{(fX)}s &=& f\nabla_Xs\, \\ \nabla_{X}(fs) &=& (Xf)s + f\nabla_Xs \,,\end{matrix}$$\begin{matrix}\nabla_{(fX)}s &=& f\nabla_Xs\, \\ \nabla_{X}(fs) &=& (Xf)s + f\nabla_Xs \,,\end{matrix}$

where $f$$f$ is a smooth function and $X$$X$ a vector field on $M$$M$ and $s$$s$ a section of $E$$E$, and where $Xf = \partial_Xf = df.X$$Xf = \partial_Xf = df.X$ is the ordinary derivative of the function $f$$f$ in the direction of $X$$X$. By these properties, $\nabla$$\nabla$ is defined locally and even pointwise regarding the first argument: For any $v\in T_pM$$v\in T_pM$ we may define $\nabla_xs := (\nabla_Xs)_p$$\nabla_xs := (\nabla_Xs)_p$ where $X$$X$ is any (local) vector field with $X_p = x$$X_p = x$.

A covariant derivative has all properties of the ordinary directional derivative for functions with exception of the commutativity: For functions $f$$f$ we have $\partial_i\partial_jf = \partial_j\partial_if$$\partial_i\partial_jf = \partial_j\partial_if$ with $\partial_if = \frac{\partial(f \circ \phi)}{\partial u_i} = \partial_{\phi_i}f$$\partial_if = \frac{\partial(f \circ \phi)}{\partial u_i} = \partial_{\phi_i}f$, where $\phi : \R^n\to M$$\phi : \R^n\to M$ is a local parametrization of $M$$M$ and $\phi_i := \partial_i\phi$$\phi_i := \partial_i\phi$ its $i$$i$-th partial derivative. Instead, for covariant derivatives, the commutator $[\nabla_i,\nabla_j]$$[\nabla_i,\nabla_j]$ (with $\nabla_i = \nabla_{\phi_i}$$\nabla_i = \nabla_{\phi_i}$) is nonzero in general, but it is only a tensor (rather than a differential operator), the curvature tensor.

Covariant derivatives are not tensors since they are derivations in the second argument, but two covariant derivatives $\nabla,\tilde\nabla$$\nabla,\tilde\nabla$ on $E$$E$ differ by a tensor: $A = \tilde\nabla-\nabla$$A = \tilde\nabla-\nabla$ defines a tensor $A \in \textup{Hom}(TM\otimes E,E)$$A \in \textup{Hom}(TM\otimes E,E)$.

2 Examples

• The Levi-Civita derivative on $TM$$TM$ where $M$$M$ is a Riemannian manifold
• The Canonical derivative on $TM$$TM$ where $M = G/H$$M = G/H$ is a reductive homogeneous space
• The Projection derivative on a subbundle $E \subset M \times \Rr^n$$E \subset M \times \Rr^n$: For any $s \in \Gamma E \subset C^\infty(M,\Rr^n)$$s \in \Gamma E \subset C^\infty(M,\Rr^n)$

and a local parametrization $\phi$$\phi$ on $M$$M$ we put

$\displaystyle \nabla_i s = (\partial_is)^E$ $$\displaystyle \nabla_i s = (\partial_is)^E$$

where $(\,\,)^E$$(\,\,)^E$ at any point $p\in M$$p\in M$ denotes the orthogonal projection from $\Rr^n$$\Rr^n$ onto the subspace $E_p$$E_p$.

For further information see [Kobayashi&Nomizu1963].

3 References

• [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