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sagemanifolds
GitHub Repository: sagemanifolds/IntroToManifolds
Path: blob/main/25Manifold_Torsion_Curvature_Forms.ipynb
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Kernel: SageMath 9.6

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).

version()
'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 {e1,...,en}\{e_1 , . . . , e_n \} of smooth vector fields defined on some open subset UU of the manifold MM such that, at each point pUp ∈ U , the tangent vectors eie_i at pp form a basis of TpMT_p M. Let the set of 1-forms {e1,...,en}\{e^1 , . . . , e^n \} be its dual basis (i.e. ei(ej)=δije^i (e_j ) = δ_{ij} ). If there exists a coordinate system (x1,...,xn)(x_1 , . . . , x_n ) such that ei=xie_i = \frac{∂}{∂ x^i} or, equivalently, ei=dxie^i = dx^i , we will say that the basis e1,...,en{e_1 , . . . , e_n } is holonomic. In the present notebook the set {e1,en}\{e_1,\ldots e_n\} will denote not necessarily holonomic basis.


Connection forms


If is a connection on a manifold M M, the connection forms, ωji\omega^i_j , with respect to the basis {e1,...,en}\{e_1 , . . . , e_n \}, are the 1-forms defined by

ωji(X)=ei(Xej),forXX(M),(25.1)\begin{darray}{rcl} \omega_j^i(X)=e^i(\nabla_Xe_j),\quad \text{for} X\in \mathfrak{X}(M), \tag{25.1} \end{darray}

or equivalently Xej=ωji(X)ei,forXX(M).(25.1’)\begin{darray}{rcl} \nabla_Xe_j=\omega_j^i(X)e_i,\quad \text{for} X\in \mathfrak{X}(M). \tag{25.1'} \end{darray}


Example 25.1

On the two-dimensional manifold we have four connection forms ωji\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]]

[ω 11,ω 21,ω 12,ω 22]\displaystyle \left[\omega^1_{\ \, 1}, \omega^1_{\ \, 2}, \omega^2_{\ \, 1}, \omega^2_{\ \, 2}\right]


If we define

Γjki=ωji(ek),(25.2)\begin{darray}{rcl} \Gamma ^i_{jk}=\omega^i_j(e_k), \tag{25.2} \end{darray}

then for XX(M)X\in \mathfrak{X}(M), we have X=ek(X)ekX=e^k(X)e_k and

ωji(X)=ωji(ek(X)ek)=ek(X)ωji(ek)=Γjkiek(X),\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., ωji=Γjkiek,(25.3)\begin{darray}{rcl} \omega_j^i=\Gamma^i_{jk}e^k, \tag{25.3} \end{darray} and according to (25.1')

eiej=ωjk(ei)ek=Γjikek,(25.4)\begin{darray}{rcl} \nabla_{e_i}e_j=\omega_j^k(e_i)e_k=\Gamma^k_{ji}e_k, \tag{25.4} \end{darray}

which is analogous to (21.5) but holds also for non-holonomic bases.


Torsion forms


If T(X,Y)=XYYX[X,Y],  X,YX(M)  T (X, Y) = ∇ X Y − ∇ Y X − [X, Y],\ \ X, Y ∈ \mathfrak{X}(M)\ \ is the torsion, then the torsion 2-forms θi\theta^i with respect to {e1,,en}\{e_1,\ldots,e_n\} are defined by

θi(X,Y)=ei(T(X,Y)),for X,YX(M),(25.5)\begin{darray}{rcl} \theta^i(X,Y)=e^i(T(X,Y)),\quad \text{for}\ X,Y\in \mathfrak{X}(M), \tag{25.5} \end{darray}

or equivalently

T(X,Y)=θi(X,Y)ei,for X,YX(M).(25.6)\begin{darray}{rcl} T(X,Y)=\theta^i(X,Y)e_i,\quad \text{for}\ X,Y\in \mathfrak{X}(M). \tag{25.6} \end{darray}

Since TT is antisymmetric, each θi\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

[θ1,θ2]\displaystyle \left[\theta^1, \theta^2\right]


First Cartan structure equation


Assume we have vector fields  X \ X\ and  Y=Yiei=ei(Y)ei. \ Y=Y^ie_i=e^i(Y)e_i.\ From (21.14) it follows

Xt(Y)=X(t(Y))t(XY),\begin{equation} \nabla_Xt(Y)=X(t(Y))-t(\nabla_XY), \tag{*} \end{equation}

for 1-forms tt on MM, so

Xei(Y)=X(ei(Y))ei(XY)=X(Yi)ei(X(Yjej))=X(Yi)ei(YjXej+(XYj)ej)=X(Yi)Yjei(Xej)ei((XYj)ej)=X(Yi)Yjei(Xej)X(Yi)=Yjωji(X)=ωji(X)ej(Y),\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 Xei=ωji(X)ej.(25.7)\begin{darray}{rcl} \nabla_Xe^i=-\omega^i_j(X)e^j. \tag{25.7} \end{darray}

Using (*) once again we obtain

ei(XY)=X(ei(Y))(Xei)(Y),ei(YX)=Y(ei(X))(Yei)(X).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

(dω)(X,Y)=Xω(Y)Yω(X)ω([X,Y]),(d\omega)(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y]),

and from the definition of the exterior product

ts=tsst.t\wedge s=t\otimes s-s\otimes t.

From those facts and (25.7) we obtain

θi(X,Y)=ei(T(X,Y))=ei(XYYX[X,Y])=X(ei(Y))(Xei)(Y)Y(ei(X))+(Yei)(X)ei([X,Y])=X(ei(Y))Y(ei(X)ei([X,Y])(Xei)(Y)+(Yei)(X)=dei(X,Y)+ωji(X)ej(Y)ωji(Y)ej(X)=(dei+ωjiej)(X,Y).\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

θi=dei+ωjiej.(25.8)\begin{darray}{rcl} \theta^i=de^i+\omega^i_j\wedge e^j. \tag{25.8} \end{darray}

Curvature forms


If   R(X,Y)=XYYX[X,Y],  X,YX(M)  \ \ R(X, Y)= ∇_X ∇_Y − ∇_Y ∇_X − ∇_{[X,Y]},\ \ X, Y ∈ \mathfrak{X}(M)\ \ is the curvature, then the curvature forms Ωji(X,Y)\Omega^i_j(X,Y) with respect to the basis {e1,e2,...,en}\{e_1 , e_2 , . . . , e_n \}, are defined by

Ωji(X,Y)=ei(R(X,Y)ej),X,YX(M),\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

R(X,Y)ej=Ωji(X,Y)ei,X,YX(M).\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

[Ω 11,Ω 21,Ω 12,Ω 22]\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)=XYYX[X,Y],Xej=ωji(X)ei,ωji(X)=ei(Xej),X(ωjk(Y)ek)=X(ωjk(Y))ek+ωjk(Y)Xek,(dω)(X,Y)=Xω(Y)Yω(X)ω([X,Y]),ts=tsst, 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 Ωji(X,Y)=ei(XYejYXej[X,Y]ej)=ei(X(ωjk(Y)ek)Y(ωjk(X)ek))ωji([X,Y])=ei(X(ωjk(Y))ek+ωjk(Y)XekY(ωjk(X))ekωjk(X)Yek)ωji([X,Y])=(X(ωjk(Y))ei(ek)+ωjk(Y)ei(Xek)Y(ωjk(X))ei(ek)ωjk(X)ei(Yek))ωji([X,Y])=X(ωji(Y))+ωjk(Y)ωki(X)Y(ωji(X))ωjk(X)ωki(Y)ωji([X,Y])=X(ωji(Y))Y(ωji(X))ωji([X,Y])+ωki(X)ωjk(Y)ωki(Y)ωjk(X)=dωji(X,Y)+(ωkiωjk)(X,Y).\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

Ωji=dωji+ωkiωjk.\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 AklA_{kl} is an antisymmetric matrix, then for 1-forms eie^i on MM and X,YX(M)X,Y\in\mathfrak{X}(M) (Aklekel)(X,Y)=Akl(ekelelek)(X,Y)=Akl(ek(X)el(Y)el(X)ek(Y)=AklXkYlAklXlYk=AklXkYl+AlkXlYk=2AklXkYl.(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 {e1,...,en}\{e_1 , . . . , e_n \} are defined by T(ei,ej)=Tijkek,R(ei,ej)ek=Rkijlel,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 θi=12Tjkiejek,Ωji=12Rjkliekel.\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 θi(X,Y)=ei(T(X,Y))=ei(T(Xkek,Ylel))=ei(XkYlTm(ek,el)em)=ei(XkYlTklmem)=XkYlTklmel(em)=XkYlTklmδmi=XkYlTkli=12(Tkliekel)(X,Y),\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 Ωji(X,Y)=ei(R(X,Y)ej)=ei(R(Xkek,Ylel)ej)=XkYlRjklmei(em)=XkYlRjklmδmi=XkYlRjkli=12(Rjkliekel)(X,Y). \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>0y>0 with the connection coefficients with respect to the bases   e1=x,  e2=y,  e1=dx,  e2=dy  \ \ e_1=\frac{\partial}{\partial x},\ \ e_2=\frac{\partial}{\partial y},\ \ e^1=dx,\ \ e^2=dy\ \ equal to Γ121=Γ211=Γ222=1y,Γ112=1y,\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)

Γ112112=1yΓ121121=1yΓ211211=1yΓ222222=1y\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 ωji\omega^i_j can be computed using the formula (25.3),   ωji=Γjkiek\ \ \omega_j^i=\Gamma^i_{jk}e^k: ω11=Γ111e1+Γ121e2=1ydy,ω21=Γ211e1+Γ221e2=1ydx,ω12=Γ112e1+Γ122e2=1ydx,ω22=Γ212e1+Γ222e2=1ydy.\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]]

[ω 11=1ydy,ω 21=1ydx,ω 12=1ydx,ω 22=1ydy]\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),   θi=dei+ωjiej\ \ \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

[θ1=0,θ2=0]\displaystyle \left[\theta^1 = 0, \theta^2 = 0\right]

The second Cartan structure equation (25.10):  Ωji=dωji+ωkiωjk\ \Omega^i_j=d\omega^i_j+\omega^i_k\wedge \omega^k_j can be used in calculation of curvature forms:

Ω11=dω11+ω11ω11+ω21ω12=d(1ydy)(1ydx)(1ydx)=1y2dydy=0,\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,

Ω21=dω21+ω11ω21+ω21ω22=d(1ydx)+(1ydy)(1ydx)+(1ydx)(1ydy)=1y2dydx+1y2dydx+1y2dxdy=1y2dxdy,\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,

Ω12=dω12+ω12ω11+ω22ω12=d(1ydx)+(1ydx)(1ydy)+(1ydy)(1ydx)=1y2dydx1y2dxdy1y2dydx=1y2dxdy,\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,

Ω22=dω22+ω12ω21+ω22ω22=d(1ydy)+(1ydx)(1ydx)=1y2dydy1y2dxdx=0.\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.
[nab.curvature_form(i,j).display() # curvature forms for i in [1,2] for j in [1,2]]

[Ω 11=0,Ω 21=1y2dxdy,Ω 12=1y2dxdy,Ω 22=0]\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) Ωji=12Rjkliekel\quad \Omega^i_j=\frac{1}{2}R^i_{jkl}e^k\wedge e^l, and RjkliR^i_{jkl} is antisymmetric in k,lk,l, then Ω21=12R2kl1ekel=12(R2121dxdy+R2211dydx)=12(R2121dxdy+R2121dxdy)=R2121dxdy=1y2dxdy,\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 Ω12\Omega^2_1, we see that the different from zero components of the curvature tensor are equal to:

R2121=1y2,  R2211=1y2,  R1122=1y2,  R1212=1y2R^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)

R12121212=1y2R12211221=1y2R21122112=1y2R21212121=1y2\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 R2R^2 with Christoffel symbols Γ111=Γ221=4u1+u2+4v2,Γ112=Γ222=4v1+u2+4v2,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]]

[ω 11=(4u4u2+4v2+1)du,ω 21=(4u4u2+4v2+1)dv,ω 12=(4v4u2+4v2+1)du,ω 22=(4v4u2+4v2+1)dv]\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:

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

[θ1=0,θ2=0]\displaystyle \left[\theta^1 = 0, \theta^2 = 0\right]

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]]

[Ω 11,Ω 21,Ω 12,Ω 22]\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.

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())

Ω 11=16uv(4u2+4v2+1)2dudv\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

Ω 21=4(4v2+1)(4u2+4v2+1)2dudv\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

Ω 12=4(4u2+1)(4u2+4v2+1)2dudv\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

Ω 22=16uv(4u2+4v2+1)2dudv\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.

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

[[[xy,x2],[x3,y2]],[[y2,xy],[x2+2y2,x2+y2]]]\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:

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

[[ω 11=xydx+x2dy,ω 21=x3dx+y2dy],[ω 12=y2dx+xydy,ω 22=(x2+2y2)dx+(x2+y2)dy]]\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)=XYYX[X,Y],forX,YX(M).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

T~(ω,X,Y)=ω(T(X,Y)),forX,YX(M),ωΩ1(M).\tilde{T}(\omega,X,Y)=\omega(T(X,Y)),\quad \text{for}\quad X,Y\in \mathfrak{X}(M), \omega\in \Omega^1(M).
print(nab.torsion()) # torsion (1,2) tensor nab.torsion().disp()
Tensor field of type (1,2) on the 2-dimensional differentiable manifold M

(x3x2)xdxdy+(x3+x2)xdydx+(x2xy+2y2)ydxdy+(x2+xy2y2)ydydx\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:

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

θ1=(x3x2)dxdy\displaystyle \theta^1 = \left( -x^{3} - x^{2} \right) \mathrm{d} x\wedge \mathrm{d} y

θ2=(x2xy+2y2)dxdy\displaystyle \theta^2 = \left( x^{2} - x y + 2 \, y^{2} \right) \mathrm{d} x\wedge \mathrm{d} y

and the curvature forms:

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

Ω 11=(x4yy4+x)dxdy\displaystyle \Omega^1_{\ \, 1} = \left( -x^{4} y - y^{4} + x \right) \mathrm{d} x\wedge \mathrm{d} y

Ω 21=(2x5+xy32y4(x3+x2)y2)dxdy\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

Ω 12=(x2y2+2xy3y4+(x31)y)dxdy\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

Ω 22=(y4+(x44)y2x)dxdy\displaystyle \Omega^2_{\ \, 2} = \left( y^{4} + {\left(x^{4} - 4\right)} y - 2 \, x \right) \mathrm{d} x\wedge \mathrm{d} y