This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
version()
To describe connections one can use bases not necessarily induced by coordinate systems. We can use sets $\{e_1 , . . . , e_n \}$ of smooth vector fields defined on some open subset $U$ of the manifold $M$ such that, at each point $p ∈ U $, the tangent vectors $e_i$ at $p$ form a basis of $T_p M$. Let the set of 1-forms $\{e^1 , . . . , e^n \}$ be its dual basis (i.e. $e^i (e_j ) = δ_{ij} $). If there exists a coordinate system $(x_1 , . . . , x_n )$ such that $e_i = \frac{∂}{∂ x^i}$ or, equivalently, $e^i = dx^i$ , we will say that the basis ${e_1 , . . . , e_n }$ is holonomic. In the present notebook the set $\{e_1,\ldots e_n\}$ will denote not necessarily holonomic basis.
If $∇$ is a connection on a manifold $ M$, the connection forms, $\omega^i_j$ , with respect to the basis $\{e_1 , . . . , e_n \}$, are the 1-forms defined by
\begin{eqnarray} \omega_j^i(X)=e^i(\nabla_Xe_j),\quad \text{for} X\in \mathfrak{X}(M), \tag{25.1} \end{eqnarray}or equivalently \begin{eqnarray} \nabla_Xe_j=\omega_j^i(X)e_i,\quad \text{for} X\in \mathfrak{X}(M). \tag{25.1'} \end{eqnarray}
Example 25.1
On the two-dimensional manifold we have four connection forms $\omega^i_j$:
%display latex
M = Manifold(2, 'M', start_index=1) # 2-dim. manifold M
c_xy.<x,y> = M.chart() # chart on M
nab = M.affine_connection('nabla', r'\nabla') # affine connection
nab[1,1,1] = 0 # all con. coeff. zero
[nab.connection_form(i,j) for i in [1,2] for j in [1,2]]
If we define
\begin{eqnarray} \Gamma ^i_{jk}=\omega^i_j(e_k), \tag{25.2} \end{eqnarray}then for $X\in \mathfrak{X}(M)$, we have $X=e^k(X)e_k$ and
$$ \omega^i_j(X)=\omega^i_j(e^k(X)e_k)=e^k(X)\omega^i_j(e_k)=\Gamma^i_{jk}e^k(X),$$i.e., \begin{eqnarray} \omega_j^i=\Gamma^i_{jk}e^k, \tag{25.3} \end{eqnarray} and according to (25.1')
\begin{eqnarray} \nabla_{e_i}e_j=\omega_j^k(e_i)e_k=\Gamma^k_{ji}e_k, \tag{25.4} \end{eqnarray}which is analogous to (21.5) but holds also for non-holonomic bases.
If $T (X, Y) = ∇ X Y − ∇ Y X − [X, Y],\ \ X, Y ∈ \mathfrak{X}(M)\ \ $ is the torsion, then the torsion 2-forms $\theta^i$ with respect to $\{e_1,\ldots,e_n\}$ are defined by
\begin{eqnarray} \theta^i(X,Y)=e^i(T(X,Y)),\quad \text{for}\ X,Y\in \mathfrak{X}(M), \tag{25.5} \end{eqnarray}or equivalently
\begin{eqnarray} T(X,Y)=\theta^i(X,Y)e_i,\quad \text{for}\ X,Y\in \mathfrak{X}(M). \tag{25.6} \end{eqnarray}Since $T$ is antisymmetric, each $\theta^i$ is 2-form.
Example 25.2
On the two-dimensional manifold we have two torsion 2-forms:
%display latex
M = Manifold(2, 'M', start_index=1) # 2-dim. manifold M
c_xy.<x,y> = M.chart() # chart on M
nab = M.affine_connection('nabla', r'\nabla') # affine connection
nab[1,1,1] = 0 # all con. coeff. zero
[nab.torsion_form(i) for i in [1,2]] # torsion 2-forms on M
Assume we have vector fields $\ X\ $ and $\ Y=Y^ie_i=e^i(Y)e_i.\ $ From (21.14) it follows
\begin{equation} \nabla_Xt(Y)=X(t(Y))-t(\nabla_XY), \tag{*} \end{equation}for 1-forms $t$ on $M$, so
$$\nabla_Xe^i(Y)=X(e^i(Y))-e^i(\nabla_XY) =X(Y^i)-e^i(\nabla_X(Y^je_j))\\ =X(Y^i)-e^i(Y^j\nabla_Xe_j+(XY^j)e_j)\\ =X(Y^i)-Y^je^i(\nabla_Xe_j)-e^i((XY^j)e_j)\\ =X(Y^i)-Y^je^i(\nabla_Xe_j)-X(Y^i)\\ =-Y^j\omega^i_j(X)=-\omega^i_j(X)e^j(Y), $$so \begin{eqnarray} \nabla_Xe^i=-\omega^i_j(X)e^j. \tag{25.7} \end{eqnarray}
Using (*) once again we obtain
$$ e^i(\nabla_XY)=X(e^i(Y))-(\nabla_Xe^i)(Y),\\ e^i(\nabla_YX)=Y(e^i(X))-(\nabla_Ye^i)(X). $$From (16.6) we know that
$$ (d\omega)(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y]), $$and from the definition of the exterior product
$$ t\wedge s=t\otimes s-s\otimes t. $$From those facts and (25.7) we obtain
$$\theta^i(X,Y)=e^i(T(X,Y))=e^i(\nabla_XY-\nabla_YX-[X,Y])\\ =X(e^i(Y))-(\nabla_Xe^i)(Y)-Y(e^i(X))+(\nabla_Ye^i)(X)-e^i([X,Y])\\ =X(e^i(Y))-Y(e^i(X)-e^i([X,Y])-(\nabla_Xe^i)(Y)+(\nabla_Ye^i)(X)\\ =de^i(X,Y)+\omega^i_j(X)e^j(Y)-\omega^i_j(Y)e^j(X)\\ =(de^i+\omega^i_j\wedge e^j)(X,Y). $$We have proved the first Cartan structural equation
\begin{eqnarray} \theta^i=de^i+\omega^i_j\wedge e^j. \tag{25.8} \end{eqnarray}If $\ \ R(X, Y)= ∇_X ∇_Y − ∇_Y ∇_X − ∇_{[X,Y]},\ \ X, Y ∈ \mathfrak{X}(M)\ \ $ is the curvature, then the curvature forms $\Omega^i_j(X,Y)$ with respect to the basis $\{e_1 , e_2 , . . . , e_n \}$, are defined by
\begin{equation} \Omega^i_j(X,Y)=e^i(R(X,Y)e_j),\quad X, Y ∈ \mathfrak{X}(M), \tag{25.9} \end{equation}or equivalently
\begin{equation} R(X,Y)e_j=\Omega^i_j(X,Y)e_i,\quad X, Y ∈ \mathfrak{X}(M). \tag{25.9'} \end{equation}Example 25.3
On the two-dimensional manifold we have four curvature forms:
%display latex
M = Manifold(2,'M',start_index=1) # manifold M
c_xy.<x,y> = M.chart() # chart on M
nab = M.affine_connection('nabla', r'\nabla') # connection
nab[1,1,1] = 0 # all coefficients zero
[nab.curvature_form(i,j) for i in [1,2] for j in [1,2]]
# curvature forms
If we use the formulas $$ R(X, Y)= ∇_X ∇_Y − ∇_Y ∇_X − ∇_{[X,Y]},\\ \nabla_Xe_j=\omega^i_j(X)e_i,\quad \omega^i_j(X)=e^i(\nabla_Xe_j),\\ \nabla_X(\omega^k_j(Y)e_k)=X(\omega^k_j(Y))e_k+\omega^k_j(Y)\nabla_Xe_k,\\ (d\omega)(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y]),\\ t\wedge s=t\otimes s-s\otimes t, $$
then we can check that $$\Omega^i_j(X,Y)=e^i(∇_X ∇_Y e_j − ∇_Y ∇_X e_j− ∇_{[X,Y]}e_j)\\ =e^i(\nabla_X(\omega^k_j(Y)e_k)-\nabla_Y(\omega^k_j(X)e_k))-\omega^i_j([X,Y])\\ =e^i(X(\omega^k_j(Y))e_k+\omega^k_j(Y)\nabla_Xe_k-Y(\omega^k_j(X))e_k-\omega^k_j(X)\nabla_Ye_k)-\omega^i_j([X,Y])\\ =(X(\omega^k_j(Y))e^i(e_k)+\omega_j^k(Y)e^i(\nabla_Xe_k)\\ -Y(\omega^k_j(X))e^i(e_k)-\omega^k_j(X)e^i(\nabla_Ye_k))- \omega^i_j([X,Y])\\ =X(\omega^i_j(Y))+\omega_j^k(Y)\omega^i_k(X)-Y(\omega^i_j(X))-\omega^k_j(X)\omega^i_k(Y)-\omega^i_j([X,Y])\\ =X(\omega^i_j(Y))-Y(\omega^i_j(X))-\omega^i_j([X,Y])+\omega_k^i(X)\omega^k_j(Y)-\omega^i_k(Y)\omega_j^k(X)\\ =d\omega^i_j(X,Y)+(\omega^i_k\wedge \omega^k_j)(X,Y). $$
We have proved the second Cartan structural equation
\begin{equation} \Omega^i_j=d\omega^i_j+\omega^i_k\wedge \omega^k_j. \tag{25.10} \end{equation}Observation: $\hskip0.5cm$ If $A_{kl}$ is an antisymmetric matrix, then for 1-forms $e^i$ on $M$ and $X,Y\in\mathfrak{X}(M)$ $$(A_{kl}e^k\wedge e^l)(X,Y)=A_{kl}(e^k\otimes e^l-e^l\otimes e^k)(X,Y)\\ =A_{kl}(e^k(X) e^l(Y)-e^l(X) e^k(Y) =A_{kl}X^kY^l-A_{kl}X^lY^k\\ =A_{kl}X^kY^l+A_{lk}X^lY^k =2A_{kl}X^kY^l. $$
If the components of the torsion and the curvature tensors with respect to the basis $\{e_1 , . . . , e_n \}$ are defined by $$T (e_i , e_j ) = T^k_{i j} e_k,\quad R(e_i , e_j )e_k = R^l_{ki j}e_l, $$ then \begin{equation} \begin{matrix} \theta^i=\frac{1}{2}T^i_{jk}e^j\wedge e^k,\\ \Omega^i_j=\frac{1}{2}R^i_{jkl}e^k\wedge e^l. \end{matrix} \tag{25.11} \end{equation}
In fact $$\theta^i(X,Y)=e^i(T(X,Y))=e^i(T(X^ke_k,Y^le_l))\\ =e^i(X^kY^lT^m(e_k,e_l)e_m)=e^i(X^kY^lT^m_{kl}e_m)\\ =X^kY^lT^m_{kl}e^l(e_m)=X^kY^lT^m_{kl}\delta^i_m=X^kY^lT^i_{kl}\\ =\frac{1}{2}(T^i_{kl}e^k\wedge e^l)(X,Y), $$ and $$ \Omega^i_j(X,Y)=e^i(R(X,Y)e_j)=e^i(R(X^ke_k,Y^le_l)e_j)\\ =X^kY^lR^m_{jkl}e^i(e_m)=X^kY^lR^m_{jkl}\delta^i_m\\ =X^kY^lR^i_{jkl}=\frac{1}{2}(R^i_{jkl}e^k\wedge e^l)(X,Y). $$
Example 25.4
Consider the half-plane $y>0$ with the connection coefficients with respect to the bases $\ \ e_1=\frac{\partial}{\partial x},\ \ e_2=\frac{\partial}{\partial y},\ \ e^1=dx,\ \ e^2=dy\ \ $ equal to $$\Gamma^1_{12}=\Gamma^1_{21}=\Gamma^2_{22}=-\frac{1}{y}, \quad\Gamma^2_{11}=\frac{1}{y},$$ and all other coefficients equal to zero.
%display latex
M = Manifold(2, 'M', start_index=1) # manifold M
c_xy.<x,y> = M.chart() # chart on M
nab = M.affine_connection('nabla', r'\nabla') # connection on M
# con. coefficients
nab[1,1,2], nab[1,2,1], nab[2,2,2], nab[2,1,1] = -1/y, -1/y, -1/y, 1/y
nab.display(coordinate_labels=False) # show conn. coeff.
# (only nonzero)
Connection forms $\omega^i_j$ can be computed using the formula (25.3), $\ \ \omega_j^i=\Gamma^i_{jk}e^k$: $$\omega^1_1=\Gamma^1_{11}e^1+\Gamma^1_{12}e^2=-\frac{1}{y}dy,\\ \omega^1_2=\Gamma^1_{21}e^1+\Gamma^1_{22}e^2=-\frac{1}{y}dx,\\ \omega^2_1=\Gamma^2_{11}e^1+\Gamma^2_{12}e^2=\frac{1}{y}dx,\\ \omega^2_2=\Gamma^2_{21}e^1+\Gamma^2_{22}e^2=-\frac{1}{y}dy. $$
[nab.connection_form(i,j).display() # connection forms
for i in [1,2] for j in [1,2]]
Torsion forms can be computed from the first Cartan structure equation (25.8), $\ \ \theta^i=de^i+\omega^i_j\wedge e^j$:
$$\theta^1=de^1+\omega^1_1\wedge e^1+\omega^1_2\wedge e^2 =d(dx)-\frac{1}{y}dy\wedge dx-\frac{1}{y}dx\wedge dy=0,\\ \theta^2=de^2+\omega^2_1\wedge e^1+\omega^2_2\wedge e^2 =d(dy)+\frac{1}{y}dx\wedge dx-\frac{1}{y}dy\wedge dy=0.\\ $$[nab.torsion_form(i).disp() for i in [1,2]] # torsion forms
The second Cartan structure equation (25.10): $\ \Omega^i_j=d\omega^i_j+\omega^i_k\wedge \omega^k_j$ can be used in calculation of curvature forms:
$$ \Omega_1^1=d\omega^1_1+\omega^1_1\wedge \omega^1_1+\omega^1_2\wedge \omega^2_1\\=d\Big(-\frac{1}{y}dy\Big)-\Big(\frac{1}{y}dx\Big)\wedge \Big(\frac{1}{y}dx\Big)\\ =\frac{1}{y^2}dy\wedge dy=0,$$[nab.curvature_form(i,j).display() # curvature forms
for i in [1,2] for j in [1,2]]
Since by (25.11) $\quad \Omega^i_j=\frac{1}{2}R^i_{jkl}e^k\wedge e^l$, and $R^i_{jkl}$ is antisymmetric in $k,l$, then $$\Omega_2^1=\frac{1}{2}R^1_{2kl}e^k\wedge e^l= \frac{1}{2}(R^1_{212}dx\wedge dy+R^1_{221}dy\wedge dx)\\ =\frac{1}{2}(R^1_{212}dx\wedge dy+R^1_{212}dx\wedge dy)=R^1_{212}dx\wedge dy=-\frac{1}{y^2}dx\wedge dy, $$
Using an analogous calculation for $\Omega^2_1$, we see that the different from zero components of the curvature tensor are equal to:
$R^1_{212} =-\frac{1}{y^2},\ \ R^1_{221} =\frac{1}{y^2},\ \ R^2_{112}=\frac{1}{y^2},\ \ R^2_{121}=-\frac{1}{y^2}$.
R = M.tensor_field(1,3,'R') # tensor field of (1,3)-type
R[:] = nab.riemann()[:] # R <- curv. (1,3) tensor f.
R.display_comp(coordinate_labels=False) # show R
#nab.riemann().display_comp(coordinate_labels=False)
Example 25.5
Consider the plane $R^2$ with Christoffel symbols $$Γ^1_{11}= Γ^1_{22}=\frac{4u}{1+u^2+4v^2},\\ Γ^2_{11}= Γ^2_{22}=\frac{4v}{1+u^2+4v^2}, $$ and the remaining symbols equal to 0.
%display latex
N = Manifold(2,name='R2',start_index=1) # manifold N
c_uv.<u,v> = N.chart() # chart on N
nab = N.affine_connection('nab1') # connection coefficients
nab[:] = [[[4*u/(4*u^2 + 4*v^2 + 1), 0], # define all coefficients
[0, 4*u/(4*u^2 + 4*v^2 + 1)]],
[[4*v/(4*u^2 + 4*v^2 + 1), 0],
[0, 4*v/(4*u^2 + 4*v^2 + 1)]]]
Compute connection forms:
[nab.connection_form(i,j).display() # connection forms
for i in [1,2] for j in [1,2]]
and torsion forms:
[nab.torsion_form(i).disp() # torsion forms
for i in [1,2]]
Now let us try to compute curvature forms.
[nab.curvature_form(i,j) # curvature forms -symbols
for i in [1,2] for j in [1,2]]
Before we show the results, we have to do some simplifications.
for i in [1,2]: # factor curvature forms
for j in [1,2]:
nab.curvature_form(i,j).comp()[:].apply_map(factor)
for i in [1,2]: # show curvature forms
for j in [1,2]:
show(nab.curvature_form(i,j).disp())
Example 25.6
In previous examples, the torsion forms were equal to zero.
Let us consider a 2-dimensional manifold with "random" connection coefficients.
M = Manifold(2, 'M', start_index=1) # manifold M
c_xy.<x,y> = M.chart() # chart on M
nab = M.affine_connection('nabla', r'\nabla') # connection on M
nab[1,1,1], nab[1,1,2] = x*y, x^2
nab[1,2,1], nab[1,2,2] = -x^3, y^2
nab[2,1,1], nab[2,1,2] = y^2, x*y
nab[2,2,1], nab[2,2,2] = x^2+y^2+y^2, y^2-x^2
%display latex # show connection
nab[:] # coefficients
Compute the connection forms:
[[nab.connection_form(i,j).display() # connection forms
for j in [1,2]] for i in [1,2]]
Recall that the torsion was defined in (21.13) as
$$T (X, Y) = ∇_X Y − ∇_Y X − [X, Y],\quad \text{for}\quad X,Y\in \mathfrak{X}(M).$$In SageMath the torsion method returns the tensor of type (1,2) defined by
print(nab.torsion()) # torsion (1,2) tensor
nab.torsion().disp()
The torsion forms in our example are equal to:
for i in [1,2]: # torsion forms
show(nab.torsion_form(i).disp())
and the curvature forms:
for i in [1,2]: # curvature forms
for j in [1,2]:
show(nab.curvature_form(i,j).disp())