common formulas in calculus of differential forms

1 Euclidean forms

To begin with we have the total differential for scalars $f:D\to ℝ$ where $D$ is a domain in ${ℝ}^n$:

$$df=\sum _s\frac{\partial f}{\partial x^s}d{x}^s$$

or by the Einstein summation convention

$$df=\frac{\partial f}{\partial x^s}d{x}^s$$

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 $d{x}^s$ are linear functionals ${ℝ}^n\to ℝ$ dual to the derivations $\frac{\partial }{\partial x^s}$, that is

$$d{x}^s\left(\frac{\partial }{\partial x^t}\right)={\delta }_t^s$$

this coincides with the calculation$d{x}^s(\frac{\partial }{\partial x^t})=\frac{\partial x^s}{\partial x^t}={\delta }_t^s$.

If $X$ is a vector field and $f$ a scalar field then one has for the directional derivative

$$Xf=X^s\frac{\partial }{\partial x^s}f=X^{s}df(\frac{\partial }{\partial x^s})=df(X)$$

For a pair of functions $g,f:D\to ℝ$ we can check Leibniz’s rule

$$d(fg)=gdf+fdg$$

Let ${\mathrm{\Omega }}^0(D)=C^{\mathrm{\infty }}(D)$ be the set of 0-forms in $D$ and let ${\mathrm{\Omega }}^1(D)=\{w=w_{s}d{x}^s:w_s\in {\mathrm{\Omega }}^0\}$(where $w_{s}d{x}^s={\sum }_s{w}_{s}d{x}^s$) be the set of 1-forms in $D$.

Then the operator $d$ can be seen as a linear operator $d:{\mathrm{\Omega }}^0(D)\to {\mathrm{\Omega }}^1(D)$.

This can be generalized by defining ${\mathrm{\Omega }}^k(D)$ to be the set of k-forms; that is,expressions of the type:

$$A_{s_1\mathrm{\dots }{s}_k}d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k}$$

where $A_{s_1\mathrm{\dots }{s}_k}$ are in ${\mathrm{\Omega }}^0(D)$ i.e. they are scalars andthey are multi-indexed sums. Further, the symbols $d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k}$ are the wedge products of the $d{x}^s$.

So $d:{\mathrm{\Omega }}^k(D)\to {\mathrm{\Omega }}^{k+1}(D)$is calculated by

$$d(A_{s_1\mathrm{\dots }{s}_k}d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k})=d(A_{s_1\mathrm{\dots }{s}_k})\wedge d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k}$$

For example, if $A=A_{s}d{x}^s$ then $dA=d{A}_s\wedge d{x}^s$, hence

$$dA=\frac{\partial A_s}{\partial x^t}d{x}^t\wedge d{x}^s$$

which is rearranged as

$$dA=\left(\frac{\partial A_s}{\partial x^t}-\frac{\partial A_t}{\partial x^s}\right)d{x}^t\wedge d{x}^s,$$

and for two forms, if $B=B_{st}d{x}^s\wedge d{x}^t$ then

$$dB=\frac{\partial B_{st}}{\partial x^u}d{x}^u\wedge d{x}^s\wedge d{x}^t.$$

Now if we have a map between two domains $F:D\to E$ and $F=(F^1,\mathrm{\dots },F^n)$, we can pullback forms as $F^{*}:{\mathrm{\Omega }}^k(E)\to {\mathrm{\Omega }}^k(D)$, beginnig with the observation that at basics $d{x}^k$, we pullback it as

$$F^{*}(d{x}^k)=d(x^k\circ F)=d{F}^k=\frac{\partial F^k}{\partial x^s}d{x}^s$$

then, if we want $\omega \mapsto F^{*}(\omega )$,where $\omega ={\omega }_{s_1\mathrm{\dots }{s}_k}d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k}$,we are going to receive

$$F^{*}(\omega )={\omega }_{s_1\mathrm{\dots }{s}_k}\circ f\frac{\partial F^{s_1}}{\partial x^{t_1}}\mathrm{\cdots }\frac{\partial F^{s_k}}{\partial x^{t_k}}d{x}^{t_1}\wedge \mathrm{\cdots }\wedge d{x}^{t_k}$$

Here the $t_i$-sumsmust be taken between all indexes obeying .

So if $\omega \in {\mathrm{\Omega }}^n(D)$, $F^{*}(\omega )={\omega }_{1\mathrm{\dots }n}\circ Fdet(F^{\prime })d{x}^1\wedge \mathrm{\cdots }\wedge d{x}^n$

We also have

$$F^{*}(v\wedge w)=F^{*}(v)\wedge F^{*}(w)$$

Obviously there are no $n+1,n+2,\mathrm{\dots }$ forms in $D$ and usually one set ${\mathrm{\Omega }}^k(D)=0$ if $k\ge n$.

2 The de Rham complex.

The collection of mappings

$$0⟶{\mathrm{\Omega }}^0(D)\stackrel{d}{⟶}{\mathrm{\Omega }}^1(D)\stackrel{d}{⟶}\mathrm{\cdots }\stackrel{d}{⟶}{\mathrm{\Omega }}^n(D)⟶0$$

give us a chain complex due that $dd=0$, so one can measure how much this differs from exactness via its homology

$$H^k(D)=\frac{\ker (d)}{\mathrm{im}(d)}$$

called the cohomological $k$-group for $D$.

Somewith the fear of being confused with the giving of the same name to the operator ${\mathrm{\Omega }}^k(D)\stackrel{d}{⟶}{\mathrm{\Omega }}^{k+1}(D)$, would like to write

$${\mathrm{\Omega }}^k(D)\stackrel{d^k}{⟶}{\mathrm{\Omega }}^{k+1}(D)$$

and then one should modify the above conventions with

$$d^{k+1}{d}^k=0$$

and

$$H^k(D)=\frac{\ker (d^k)}{\mathrm{im}(d^{k-1})}$$

3 Manifold’s Forms.

One had seen that for mappings $F:D\to E$ between ${ℝ}^n$’s domains behave as $F^{*}:{\mathrm{\Omega }}^k(E)\to {\mathrm{\Omega }}^k(D)$.Then we can assign k-forms in each chart $(U,\mathrm{\Phi })$ of a n-manifold $M$by means of the coordinated functions $u^i=x^i\circ \mathrm{\Phi }$ on the neighborhood $U$.Then

$$d{u}^i=d(x^i\circ \mathrm{\Phi })={\mathrm{\Phi }}^{*}d{x}^i$$

which will be the duals of the derivations$\frac{\partial }{\partial u^j}$.

Observe that if ${\mathrm{\Phi }}^{*}:{\mathrm{\Omega }}^0(\varphi (U))\to {\mathrm{\Omega }}^0(U)$then $\mathrm{\Phi }(g)=g\circ \mathrm{\Phi }$ is a scalar in $U$.

If ${\mathrm{\Phi }}^{*}:{\mathrm{\Omega }}^1(\varphi (U))\to {\mathrm{\Omega }}^1(U)$ then

$${\mathrm{\Phi }}^{*}(w_{s}d{x}^s)=w_s\circ \mathrm{\Phi }{\mathrm{\Phi }}^{*}(d{x}^s)=w_s\circ \mathrm{\Phi }d{u}^s$$

For $k$-forms

$$w_{s_1{s}_2\mathrm{\dots }{s}_k}d{u}^{s_1}\wedge \mathrm{\cdots }\wedge d{u}^{s_k}=w_{s_1{s}_2\mathrm{\dots }{s}_k}\circ {\mathrm{\Phi }}^{-1}\circ \mathrm{\Phi }d(x^{s_1}\circ \mathrm{\Phi })\wedge \mathrm{\cdots }\wedge d(x^{s_k}\circ \mathrm{\Phi })$$

$$={\mathrm{\Phi }}^{*}(w_{s_1{s}_2\mathrm{\dots }{s}_k}\circ {\mathrm{\Phi }}^{-1}){\mathrm{\Phi }}^{*}(d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k})$$

$$={\mathrm{\Phi }}^{*}(w_{s_1{s}_2\mathrm{\dots }{s}_k}\circ {\mathrm{\Phi }}^{-1}d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k})$$

where $w_{s_1{s}_2\mathrm{\dots }{s}_k}\circ {\mathrm{\Phi }}^{-1}d{x}^{s_1}\wedge \mathrm{\cdots }\wedge d{x}^{s_k}$ is a $k$-form in $\mathrm{\Phi }(U)$.

4 Forms and connections

A connection is a bi-linear operator $\nabla :\mathrm{\Gamma }{(TM)}^2\to \mathrm{\Gamma }(TM)$ where $\mathrm{\Gamma }(TM)$ is the space of differentiable sections in the tangent bundle.

The Chistoffel symbols ${\mathrm{\Gamma }}_{ij}^s$ are the components of ${\nabla }_{{\partial }_i}{\partial }_j$ through the equation

$${\nabla }_{{\partial }_i}{\partial }_j={\mathrm{\Gamma }}_{ij}^s{\partial }_s$$

where the ${\partial }_s$ are the coordinated tangent vectors.

The curvature tensor is defined as

$$R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$$

which is a tri-linear map $\mathrm{\Gamma }{(TM)}^3\to \mathrm{\Gamma }(TM)$, 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

$$R(X,Y){\partial }_j=\mathrm{\Omega }^s{}_j(X,Y){\partial }_s$$

these $\omega ^s{}_j$ and $\mathrm{\Omega }^s{}_j$ define a 1-form and a 2-form viewed as a sections $M\to {\mathrm{\Omega }}^1(TM)$ and $M\to {\mathrm{\Omega }}^2(TM)$ respectively.

Observe that ${\nabla }_{{\partial }_k}{\partial }_j=\omega ^s{}_j({\partial }_k){\partial }_s$ which compared with ${\nabla }_{{\partial }_k}{\partial }_j={\mathrm{\Gamma }}_{kj}^s{\partial }_s$, it implies $\omega ^s{}_j({\partial }_k)={\mathrm{\Gamma }}_{kj}^s$and for an arbitrary vector field $X=X^k{\partial }_k$ (in the tangent coordinated basis)

$$\omega ^s{}_j(X)=X^k{\mathrm{\Gamma }}_{kj}^s$$

Let $X_1,X_2,\mathrm{\dots },X_n$ 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 $T_{p}M$.

Define thru

$${\nabla }_{X_i}{X}_j={\widehat{\mathrm{\Gamma }}}_{ij}^s{X}_s$$

a an-holonomic connection coefficients

and

$$R(X_i,X_j)X_k={\widehat{R^s}}_{ijk}{X}_s$$

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

$$c^s{}_{ij}={\widehat{\mathrm{\Gamma }}}_{ij}^s-{\widehat{\mathrm{\Gamma }}}_{ji}^s$$

5 Cartan Structural Equations

The connection and the curvature forms satisfy the premiere$d{\theta }^i=-{\widehat{{\omega }^i}}_s\wedge {\theta }^s$,where the ${\theta }^i$ are the 1-forms dual to the $X_j$ and the deuxieme${\widehat{{\mathrm{\Omega }}^i}}_j=d{\widehat{{\omega }^i}}_j+{\widehat{{\omega }^i}}_s\wedge {\widehat{{\omega }^s}}_j$where the corresponding connection forms are calculated by${\nabla }_Y{X}_j={\widehat{{\omega }^s}}_j(Y)X_s$i.e.

$${\widehat{{\omega }^l}}_j={\widehat{\mathrm{\Gamma }}}_{js}^l{\theta }_s.$$

All that fits perfectly to give

$${\widehat{{\mathrm{\Omega }}^i}}_j=\frac{1}{2}{\widehat{R^i}}_{jkl}{\theta }^k\wedge {\theta }^l$$

with .

This shows that the calculations of ${\widehat{R^i}}_{jkl}$ are very easy objects to put into an algorithm (Debever).