Notebook

25. Torsion and curvature forms¶

This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).

In [1]:

version()

Out[1]:

'SageMath version 9.6, Release Date: 2022-05-15'

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.

Connection forms¶

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 $\omega^i_j$:

In [2]:

%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]]

Out[2]:

$\displaystyle \left[\omega^1_{\ \, 1}, \omega^1_{\ \, 2}, \omega^2_{\ \, 1}, \omega^2_{\ \, 2}\right]$

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.

Torsion forms¶

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:

In [3]:

%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

Out[3]:

$\displaystyle \left[\theta^1, \theta^2\right]$

First Cartan structure equation¶

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}

Curvature forms¶

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:

In [4]:

%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

Out[4]:

$\displaystyle \left[\Omega^1_{\ \, 1}, \Omega^1_{\ \, 2}, \Omega^2_{\ \, 1}, \Omega^2_{\ \, 2}\right]$

Second Cartan structural equations¶

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.

In [5]:

%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)

Out[5]:

$\displaystyle \begin{array}{lcl} \Gamma_{\phantom{\, 1}\,1\,2}^{\,1\phantom{\, 1}\phantom{\, 2}} & = & -\frac{1}{y} \\ \Gamma_{\phantom{\, 1}\,2\,1}^{\,1\phantom{\, 2}\phantom{\, 1}} & = & -\frac{1}{y} \\ \Gamma_{\phantom{\, 2}\,1\,1}^{\,2\phantom{\, 1}\phantom{\, 1}} & = & \frac{1}{y} \\ \Gamma_{\phantom{\, 2}\,2\,2}^{\,2\phantom{\, 2}\phantom{\, 2}} & = & -\frac{1}{y} \end{array}$

Connection forms $\omega^i_j$ can be computed using the formula (25.3), $\ \ \omega_j^i=\Gamma^i_{jk}e^k$: $$\omega^1_1=\Gamma^1_{11}e^1+\Gamma^1_{12}e^2=-\frac{1}{y}dy,\\ \omega^1_2=\Gamma^1_{21}e^1+\Gamma^1_{22}e^2=-\frac{1}{y}dx,\\ \omega^2_1=\Gamma^2_{11}e^1+\Gamma^2_{12}e^2=\frac{1}{y}dx,\\ \omega^2_2=\Gamma^2_{21}e^1+\Gamma^2_{22}e^2=-\frac{1}{y}dy. $$

In [6]:

[nab.connection_form(i,j).display()             # connection forms
 for i in [1,2] for j in [1,2]]

Out[6]:

$\displaystyle \left[\omega^1_{\ \, 1} = -\frac{1}{y} \mathrm{d} y, \omega^1_{\ \, 2} = -\frac{1}{y} \mathrm{d} x, \omega^2_{\ \, 1} = \frac{1}{y} \mathrm{d} x, \omega^2_{\ \, 2} = -\frac{1}{y} \mathrm{d} y\right]$

Torsion forms can be computed from the first Cartan structure equation (25.8), $\ \ \theta^i=de^i+\omega^i_j\wedge e^j$:

$$\theta^1=de^1+\omega^1_1\wedge e^1+\omega^1_2\wedge e^2 =d(dx)-\frac{1}{y}dy\wedge dx-\frac{1}{y}dx\wedge dy=0,\\ \theta^2=de^2+\omega^2_1\wedge e^1+\omega^2_2\wedge e^2 =d(dy)+\frac{1}{y}dx\wedge dx-\frac{1}{y}dy\wedge dy=0.\\ $$

In [7]:

[nab.torsion_form(i).disp() for i in [1,2]]    # torsion forms

Out[7]:

$\displaystyle \left[\theta^1 = 0, \theta^2 = 0\right]$

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:

$$ \Omega_1^1=d\omega^1_1+\omega^1_1\wedge \omega^1_1+\omega^1_2\wedge \omega^2_1\\=d\Big(-\frac{1}{y}dy\Big)-\Big(\frac{1}{y}dx\Big)\wedge \Big(\frac{1}{y}dx\Big)\\ =\frac{1}{y^2}dy\wedge dy=0,$$
$$ \Omega^1_2=d\omega^1_2+\omega^1_1\wedge \omega^1_2+\omega^1_2\wedge \omega^2_2\\ =d\Big(-\frac{1}{y}dx\Big) +\Big(-\frac{1}{y}dy\Big)\wedge \Big(-\frac{1}{y}dx\Big) +\Big(-\frac{1}{y}dx\Big)\wedge \Big(-\frac{1}{y}dy\Big)\\ =\frac{1}{y^2}dy\wedge dx+\frac{1}{y^2}dy\wedge dx+\frac{1}{y^2}dx\wedge dy=-\frac{1}{y^2}dx\wedge dy,$$
$$ \Omega^2_1=d\omega^2_1+\omega^2_1\wedge \omega^1_1+\omega^2_2\wedge \omega^2_1\\ =d\Big(-\frac{1}{y}dx\Big) +\Big(\frac{1}{y}dx\Big)\wedge \Big(-\frac{1}{y}dy\Big) +\Big(-\frac{1}{y}dy\Big)\wedge \Big(\frac{1}{y}dx\Big)\\ =\frac{1}{y^2}dy\wedge dx-\frac{1}{y^2}dx\wedge dy-\frac{1}{y^2}dy\wedge dx=\frac{1}{y^2}dx\wedge dy, $$
$$ \Omega^2_2=d\omega^2_2+\omega^2_1\wedge \omega^1_2+\omega^2_2\wedge \omega^2_2\\ =d\Big(-\frac{1}{y}dy\Big)+ \Big(\frac{1}{y}dx\Big)\wedge \Big(-\frac{1}{y}dx\Big)\\ =\frac{1}{y^2}dy\wedge dy-\frac{1}{y^2}dx\wedge dx=0. $$

In [8]:

[nab.curvature_form(i,j).display()            # curvature forms
   for i in [1,2] for j in [1,2]]

Out[8]:

$\displaystyle \left[\Omega^1_{\ \, 1} = 0, \Omega^1_{\ \, 2} = -\frac{1}{y^{2}} \mathrm{d} x\wedge \mathrm{d} y, \Omega^2_{\ \, 1} = \frac{1}{y^{2}} \mathrm{d} x\wedge \mathrm{d} y, \Omega^2_{\ \, 2} = 0\right]$

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:

$R^1_{212} =-\frac{1}{y^2},\ \ R^1_{221} =\frac{1}{y^2},\ \ R^2_{112}=\frac{1}{y^2},\ \ R^2_{121}=-\frac{1}{y^2}$.

In [9]:

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)

Out[9]:

$\displaystyle \begin{array}{lcl} R_{\phantom{\, 1}\,2\,1\,2}^{\,1\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}} & = & -\frac{1}{y^{2}} \\ R_{\phantom{\, 1}\,2\,2\,1}^{\,1\phantom{\, 2}\phantom{\, 2}\phantom{\, 1}} & = & \frac{1}{y^{2}} \\ R_{\phantom{\, 2}\,1\,1\,2}^{\,2\phantom{\, 1}\phantom{\, 1}\phantom{\, 2}} & = & \frac{1}{y^{2}} \\ R_{\phantom{\, 2}\,1\,2\,1}^{\,2\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}} & = & -\frac{1}{y^{2}} \end{array}$

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.

In [10]:

%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:

In [11]:

[nab.connection_form(i,j).display()      # connection forms
   for i in [1,2] for j in [1,2]]

Out[11]:

$\displaystyle \left[\omega^1_{\ \, 1} = \left( \frac{4 \, u}{4 \, u^{2} + 4 \, v^{2} + 1} \right) \mathrm{d} u, \omega^1_{\ \, 2} = \left( \frac{4 \, u}{4 \, u^{2} + 4 \, v^{2} + 1} \right) \mathrm{d} v, \omega^2_{\ \, 1} = \left( \frac{4 \, v}{4 \, u^{2} + 4 \, v^{2} + 1} \right) \mathrm{d} u, \omega^2_{\ \, 2} = \left( \frac{4 \, v}{4 \, u^{2} + 4 \, v^{2} + 1} \right) \mathrm{d} v\right]$

and torsion forms:

In [12]:

[nab.torsion_form(i).disp()             # torsion forms
     for i in [1,2]]

Out[12]:

$\displaystyle \left[\theta^1 = 0, \theta^2 = 0\right]$

Now let us try to compute curvature forms.

In [13]:

[nab.curvature_form(i,j)               # curvature forms -symbols
 for i in [1,2] for j in [1,2]]

Out[13]:

$\displaystyle \left[\Omega^1_{\ \, 1}, \Omega^1_{\ \, 2}, \Omega^2_{\ \, 1}, \Omega^2_{\ \, 2}\right]$

Before we show the results, we have to do some simplifications.

In [14]:

for i in [1,2]:                       # factor curvature forms
    for j in [1,2]:
        nab.curvature_form(i,j).comp()[:].apply_map(factor)

In [15]:

for i in [1,2]:                       # show curvature forms
    for j in [1,2]:
        show(nab.curvature_form(i,j).disp())

$\displaystyle \Omega^1_{\ \, 1} = \frac{16 \, u v}{{\left(4 \, u^{2} + 4 \, v^{2} + 1\right)}^{2}} \mathrm{d} u\wedge \mathrm{d} v$

$\displaystyle \Omega^1_{\ \, 2} = \frac{4 \, {\left(4 \, v^{2} + 1\right)}}{{\left(4 \, u^{2} + 4 \, v^{2} + 1\right)}^{2}} \mathrm{d} u\wedge \mathrm{d} v$

$\displaystyle \Omega^2_{\ \, 1} = -\frac{4 \, {\left(4 \, u^{2} + 1\right)}}{{\left(4 \, u^{2} + 4 \, v^{2} + 1\right)}^{2}} \mathrm{d} u\wedge \mathrm{d} v$

$\displaystyle \Omega^2_{\ \, 2} = -\frac{16 \, u v}{{\left(4 \, u^{2} + 4 \, v^{2} + 1\right)}^{2}} \mathrm{d} u\wedge \mathrm{d} v$

Example 25.6

In previous examples, the torsion forms were equal to zero.

Let us consider a 2-dimensional manifold with "random" connection coefficients.

In [16]:

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

In [17]:

%display latex                                   # show connection
nab[:]                                           # coefficients

Out[17]:

$\displaystyle \left[\left[\left[x y, x^{2}\right], \left[-x^{3}, y^{2}\right]\right], \left[\left[y^{2}, x y\right], \left[x^{2} + 2 \, y^{2}, -x^{2} + y^{2}\right]\right]\right]$

Compute the connection forms:

In [18]:

[[nab.connection_form(i,j).display()             # connection forms
  for j in [1,2]] for i in [1,2]]

Out[18]:

$\displaystyle \left[\left[\omega^1_{\ \, 1} = x y \mathrm{d} x + x^{2} \mathrm{d} y, \omega^1_{\ \, 2} = -x^{3} \mathrm{d} x + y^{2} \mathrm{d} y\right], \left[\omega^2_{\ \, 1} = y^{2} \mathrm{d} x + x y \mathrm{d} y, \omega^2_{\ \, 2} = \left( x^{2} + 2 \, y^{2} \right) \mathrm{d} x + \left( -x^{2} + y^{2} \right) \mathrm{d} y\right]\right]$

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

$$\tilde{T}(\omega,X,Y)=\omega(T(X,Y)),\quad \text{for}\quad X,Y\in \mathfrak{X}(M), \omega\in \Omega^1(M). $$

In [19]:

print(nab.torsion())                          # torsion (1,2) tensor
nab.torsion().disp()

Tensor field of type (1,2) on the 2-dimensional differentiable manifold M

Out[19]:

$\displaystyle \left( -x^{3} - x^{2} \right) \frac{\partial}{\partial x }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( x^{3} + x^{2} \right) \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} x + \left( x^{2} - x y + 2 \, y^{2} \right) \frac{\partial}{\partial y }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( -x^{2} + x y - 2 \, y^{2} \right) \frac{\partial}{\partial y }\otimes \mathrm{d} y\otimes \mathrm{d} x$

The torsion forms in our example are equal to:

In [20]:

for i in [1,2]:                             # torsion forms
    show(nab.torsion_form(i).disp())

$\displaystyle \theta^1 = \left( -x^{3} - x^{2} \right) \mathrm{d} x\wedge \mathrm{d} y$

$\displaystyle \theta^2 = \left( x^{2} - x y + 2 \, y^{2} \right) \mathrm{d} x\wedge \mathrm{d} y$

and the curvature forms:

In [21]:

for i in [1,2]:                            # curvature forms
    for j in [1,2]:
        show(nab.curvature_form(i,j).disp())

$\displaystyle \Omega^1_{\ \, 1} = \left( -x^{4} y - y^{4} + x \right) \mathrm{d} x\wedge \mathrm{d} y$

$\displaystyle \Omega^1_{\ \, 2} = \left( 2 \, x^{5} + x y^{3} - 2 \, y^{4} - {\left(x^{3} + x^{2}\right)} y^{2} \right) \mathrm{d} x\wedge \mathrm{d} y$

$\displaystyle \Omega^2_{\ \, 1} = \left( x^{2} y^{2} + 2 \, x y^{3} - y^{4} + {\left(x^{3} - 1\right)} y \right) \mathrm{d} x\wedge \mathrm{d} y$

$\displaystyle \Omega^2_{\ \, 2} = \left( y^{4} + {\left(x^{4} - 4\right)} y - 2 \, x \right) \mathrm{d} x\wedge \mathrm{d} y$