|
To begin with we have the total differential for scalars $f\colon D\to \mathbb{R}$ where $D$ is a domain in $\mathbb{R}^n$ : $$df=\sum_s\frac{\partial f}{\partial x^s}dx^s$$ or by the Einstein summation convention $$df=\frac{\partial f}{\partial x^s}dx^s$$ which are a special case of the so-called Euclidean 1-forms. Here we reconize the covariant form of the gradient of $f$ in contravaiant ''state'': $$\nabla f=\frac{\partial f}{\partial x^s}$$ being the components of $df$ .
Here the symbols $dx^s$ are linear functionals $\mathbb{R}^n\to \mathbb{R}$ dual to the derivations $\frac{\partial }{\partial x^s}$ , that is $$dx^s\Big(\frac{\partial}{\partial x^t}\Big)=\delta^s_t$$ this coincides with the calculation $dx^s(\frac{\partial }{\partial x^t})=\frac{\partial x^s}{\partial x^t}=\delta^s_t$ .
If $X$ is a vector field and $f$ a scalar field then one has for the directional derivative $$Xf=X^s\frac{\partial}{\partial x^s}f=X^sdf(\frac{\partial}{\partial x^s})=df(X)$$
For a pair of functions $g,f\colon D\to \mathbb{R}$ we can check Leibniz's rule $$d(fg)=gdf+fdg$$
Let $\Omega^0(D)=C^{\infty}(D)$ be the set of 0-forms in $D$ and let $\Omega^1(D)=\{ w=w_sdx^s\colon w_s\in\Omega^0\}$ (where $w_sdx^s=\sum_sw_sdx^s$ ) be the set of 1-forms in $D$ .
Then the operator $d$ can be seen as a linear operator $d\colon \Omega^0(D)\to\Omega^1(D)$ .
This can be generalized by defining $\Omega^k(D)$ to be the set of k-forms; that is, expressions of the type: $$A_{s_1...s_k}dx^{s_1}\wedge\cdots\wedge dx^{s_k}$$ where $A_{s_1...s_k}$ are in $\Omega^0(D)$ i.e. they are scalars and they are multi-indexed sums. Further, the symbols $dx^{s_1}\wedge\cdots\wedge dx^{s_k}$ are the wedge products of the $dx^s$ .
So $d\colon \Omega^k(D)\to\Omega^{k+1}(D)$ is calculated by $$d(A_{s_1...s_k}dx^{s_1}\wedge\cdots\wedge dx^{s_k})= d(A_{s_1...s_k})\wedge dx^{s_1}\wedge\cdots\wedge dx^{s_k}$$
For example, if $A=A_sdx^s$ then $dA=dA_s\wedge dx^s$ , hence $$dA=\frac{\partial A_s}{\partial x^t}dx^t\wedge dx^s$$ which is rearranged as
$$dA=\Big(\frac{\partial A_s}{\partial x^t}-\frac{\partial A_t}{\partial x^s}\Big)dx^t\wedge dx^s,$$
and for two forms, if $B=B_{st}dx^s\wedge dx^t$ then $$dB=\frac{\partial B_{st}}{\partial x^u}dx^u\wedge dx^s\wedge dx^t.$$
Now if we have a map between two domains $F\colon D\to E$ and $F=(F^1,...,F^n)$ , we can pullback forms as $F^*\colon \Omega^k(E)\to \Omega^k(D)$ , beginnig with the observation that at basics $dx^k$ , we pullback it as $$F^*(dx^k)=d(x^k\circ F)=dF^k=\frac{\partial F^k}{\partial x^s}dx^s$$ then, if we want $\omega\mapsto F^*(\omega)$ , where $\omega=\omega_{s_1...s_k}dx^{s_1}\wedge\cdots\wedge dx^{s_k}$ , we are going to receive $$F^*(\omega)=\omega_{s_1...s_k}\circ f\ \frac{\partial F^{s_1}}{\partial x^{t_1}}\cdots\frac{\partial F^{s_k}}{\partial x^{t_k}}dx^{t_1}\wedge\cdots \wedge dx^{t_k}$$ Here the $t_i$ -sums must be taken between all indexes obeying $1\le t_1< t_2<\cdots < t_k\le n$ .
So if $\omega\in\Omega^n(D)$ , $F^*(\omega)=\omega_{1...n}\circ F\ \det(F') dx^1\wedge\cdots\wedge dx^n$
We also have $$F^*(v\wedge w)=F^*(v)\wedge F^*(w)$$
Obviously there are no $n+1, n+2,\ldots$ forms in $D$ and usually one set $\Omega^k(D)=0$ if $k\ge n$ .
The collection of mappings $$0\longrightarrow\Omega^0(D)\stackrel{d}\longrightarrow\Omega^1(D)\stackrel{d} \longrightarrow\cdots\stackrel{d}\longrightarrow\Omega^n(D)\longrightarrow 0$$ give us a chain complex due that $dd=0$ , so one can measure how much this differs from exactness via its homology $$H^k(D)=\frac{\operatorname{ker}(d)}{\operatorname{im}(d)}$$ called the cohomological $k$ -group for
$D$ .
Some with the fear of being confused with the giving of the same name to the operator $\Omega^k(D)\stackrel{d}\longrightarrow\Omega^{k+1}(D)$ , would like to write $$\Omega^k(D)\stackrel{d^k}\longrightarrow\Omega^{k+1}(D)$$ and then one should modify the above conventions with $$d^{k+1}d^k=0$$ and $$H^k(D)=\frac{\operatorname{ker}(d^k)}{\operatorname{im}(d^{k-1})}$$
One had seen that for mappings $F\colon D\to E$ between $\mathbb{R}^n$ 's domains behave as $F^*\colon \Omega^k(E)\to\Omega^k(D)$ . Then we can assign k-forms in each chart $(U,\Phi)$ of a n-manifold $M$ by means of the coordinated functions $u^i=x^i\circ\Phi$ on the neighborhood $U$ . Then $$du^i=d(x^i\circ\Phi)=\Phi^*dx^i$$ which will be the duals of the derivations $\frac{\partial}{\partial u^j}$ .
Observe that if $\Phi^*\colon\Omega^0(\phi(U))\to\Omega^0(U)$ then $\Phi(g)=g\circ\Phi$ is a scalar in $U$ .
If $\Phi^*\colon\Omega^1(\phi(U))\to\Omega^1(U)$ then $$\Phi^*(w_sdx^s)=w_s\circ\Phi \Phi^*(dx^s)=w_s\circ\Phi du^s$$
For $k$ -forms $$ w_{s_1s_2...s_k}du^{s_1}\wedge\cdots \wedge du^{s_k}= w_{s_1s_2...s_k}\circ\Phi^{-1}\circ\Phi d(x^{s_1}\circ\Phi)\wedge\cdots \wedge d(x^{s_k}\circ\Phi)$$
$$=\Phi^*(w_{s_1s_2...s_k}\circ\Phi^{-1}) \Phi^*(dx^{s_1}\wedge\cdots\wedge dx^{s_k})$$ $$=\Phi^*(w_{s_1s_2...s_k}\circ\Phi^{-1}dx^{s_1}\wedge\cdots\wedge dx^{s_k})$$ where $w_{s_1s_2...s_k}\circ\Phi^{-1}dx^{s_1}\wedge\cdots\wedge dx^{s_k}$ is a $k$ -form in $\Phi(U)$ .
A connection is a bi-linear operator $\nabla:\Gamma(TM)^2\to \Gamma(TM)$ where $\Gamma(TM)$ is the space of differentiable sections in the tangent bundle.
The Chistoffel symbols $\Gamma^s_{ij}$ are the components of $\nabla_{\partial_i}\partial_j$ through the equation $$\nabla_{\partial_i}\partial_j=\Gamma^s_{ij}\partial_s$$ where the $\partial_s$ are the coordinated tangent vectors.
The curvature tensor is defined as $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ which is a tri-linear map $\Gamma(TM)^3\to \Gamma(TM)$ , so the Riemann-Chistoffel symbols are defined by the components ${R^s}_{ijk}$ of $$R(\partial_i,\partial_j)\partial_k={R^s}_{ijk}\partial_s$$
With these one define the connection forms and the curvature forms as $$\nabla_X\partial_j={\omega^s}_j(X)\partial_s$$ and $$R(X,Y)\partial_j={\Omega^s}_j(X,Y)\partial_s$$ these ${\omega^s}_j$ and ${\Omega^s}_j$ define a 1-form and a 2-form viewed as a sections $M\to \Omega^1(TM)$ and $M\to \Omega^2(TM)$ respectively.
Observe that $\nabla_{\partial_k}\partial_j={\omega^s}_j(\partial_k)\partial_s$ which compared with $\nabla_{\partial_k}\partial_j=\Gamma^s_{kj}\partial_s$ , it implies ${\omega^s}_j(\partial_k)=\Gamma^s_{kj}$ and for an arbitrary vector field $X=X^k\partial_k$ (in the tangent coordinated basis) $${\omega^s}_j(X)=X^k\Gamma^s_{kj}$$
Let $X_1,X_2,...,X_n$ be another frame field (the $\partial_i$ are the coordinated frame field) , i.e. a system of $n$ -tangent vectors which are linearly independent in the tangent space, i.e, they span each $T_pM$ .
Define thru $$\nabla_{X_i}X_j=\hat{\Gamma}^s_{ij}X_s$$ a an-holonomic connection coefficients
and
$$R(X_i,X_j)X_k={ {\hat{R^s}} }_{ijk} X_s$$ as the an-holonomic.
Remember that in the coordinated frame field $[\partial_i,\partial_j]=0$ , but since $\nabla_{X_i}X_j-\nabla_{X_j}X_i=[X_i,X_j]$ this define the structural "constants" $${c^s}_{ij}X_s=[X_i,X_j]$$ and the give relation $${c^s}_{ij}=\hat{\Gamma}^s_{ij}-\hat{\Gamma}^s_{ji}$$
The connection and the curvature forms satisfy the premiere $d\theta^i=- {\hat{\omega^i}}_s\wedge\theta^s$ , where the $\theta^i$ are the 1-forms dual to the $X_j$ and the deuxieme ${\hat{\Omega^i}}_j=d{{ \hat{\omega^i}} }_j+{{ \hat{\omega^i}} }_s\wedge {{\hat{\omega^s}} }_j$ where the corresponding connection forms are calculated by $\nabla_YX_j=\hat{\omega^s}_j(Y)X_s$ i.e. $$\hat{\omega^l}_j=\hat{\Gamma}^l_{js}\theta_s.$$ All that fits perfectly to give $$\hat{\Omega^i}_j={1\over2}\hat{R^i}_{jkl} \theta^k \wedge \theta^l$$ with $k<l$ .
This shows that the calculations of $\hat{R^i}_{jkl}$ are very easy objects to put into an algorithm (Debever).
| 
