Curvature tensor and second 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$ a vector bundle over $M$$M$ with covariant derivative $\nabla$$\nabla$. Let $\phi : \R^n_o \to M$$\phi : \R^n_o \to M$ be a local parametrization of $M$$M$, defined on some open subset $\R^n_o \subset \R^n$$\R^n_o \subset \R^n$ and let $\nabla_i = \nabla_{\phi_i}$$\nabla_i = \nabla_{\phi_i}$ where $\phi_i = \partial_i\phi$$\phi_i = \partial_i\phi$ are the coordinate vector fields. The commutator $[\nabla_i,\nabla_j]$$[\nabla_i,\nabla_j]$ is in general nonzero, but it is no longer a differential operator, but a tensor $R_{ij}\in \textup{Hom}(E,E)$$R_{ij}\in \textup{Hom}(E,E)$,

(1)$R_{ij}s := [\nabla_i,\nabla_j]s = \nabla_i\nabla_js - \nabla_j\nabla_is$$R_{ij}s := [\nabla_i,\nabla_j]s = \nabla_i\nabla_js - \nabla_j\nabla_is$

for all $s\in \Gamma E$$s\in \Gamma E$. More generally, for arbitrary vector fields $X,Y$$X,Y$ with $X \circ \phi = \sum_i\xi^i\phi_i$$X \circ \phi = \sum_i\xi^i\phi_i$ and $Y \circ \phi = \sum_j\eta^j\phi_j$$Y \circ \phi = \sum_j\eta^j\phi_j$ we put

(2)$R(X,Y)s = \sum_{ij} \xi^i\eta^j R_{ij}s = [\nabla_X,\nabla_Y]s - \nabla_{[X,Y]}s$$R(X,Y)s = \sum_{ij} \xi^i\eta^j R_{ij}s = [\nabla_X,\nabla_Y]s - \nabla_{[X,Y]}s$

where $[X,Y] = \sum_i \left((X\eta^i) - (Y\xi^i)\right)\phi_i$$[X,Y] = \sum_i \left((X\eta^i) - (Y\xi^i)\right)\phi_i$ denotes the Lie bracket of vector fields. This defines a tensor $R \in \textup{Hom}(TM\otimes TM\otimes E,E)$$R \in \textup{Hom}(TM\otimes TM\otimes E,E)$, called curvature tensor.

If in addition we have any connection $\nabla$$\nabla$ on $TM$$TM$ which is torsion free, we may view $R$$R$ as the antisymmetric part of the second derivative of sections as follows. The covariant derivative of any section $s \in \Gamma E$$s \in \Gamma E$ is a tensor $\nabla s \in \Gamma\textup{Hom}(TM,E)$$\nabla s \in \Gamma\textup{Hom}(TM,E)$ which has again a covariant derivative (tensor derivative) $\nabla^2s$$\nabla^2s$. This defines a tensor $\nabla^2s \in \Gamma\textup{Hom}(TM\otimes TM,E)$$\nabla^2s \in \Gamma\textup{Hom}(TM\otimes TM,E)$, the second covariant derivative of $s$$s$, with

(3)$\nabla^2_{X,Y}s = \nabla_X\nabla_Ys - \nabla_{\nabla_XY}s$$\nabla^2_{X,Y}s = \nabla_X\nabla_Ys - \nabla_{\nabla_XY}s$

for all $X,Y \in \Gamma TM$$X,Y \in \Gamma TM$. Since $\nabla_XY -\nabla_YX = [X,Y]$$\nabla_XY -\nabla_YX = [X,Y]$, we obtain

(4)$R(X,Y)s = \nabla^2_{X,Y}s - \nabla^2_{Y,X}s.$$R(X,Y)s = \nabla^2_{X,Y}s - \nabla^2_{Y,X}s.$