24Manifold_RiemannCurvature.ipynb

Kernel: SageMath 9.6

24. Riemannian curvature tensor of type (0,4)

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'

Assume now that $M$ is a Riemannian manifold with metric $g$ and $\nabla$ is the corresponding Levi-Civita connection. We define Riemannian curvature tensor of type $(0,4)$ by

$$\begin{equation} R(X, Y, Z , W ) = g(R(X, Y )Z , W ),\quad \text{for } X,Y,Z,W\in \mathfrak{X}(M). \tag{24.1} \end{equation}$$

Since $g$ is non-singular (cf. notebook 22) the curvature (0,4)-type tensor contains the same information as the corresponding (1,3)-type tensor from the previous notebook.

In a local coordinate system $(x^1,\ldots,x^n)$ we have

$$\begin{equation} R_{ijkl}=g_{im}R^m_{jkl}. \tag{24.1'} \end{equation}$$

Symmetries of Riemannian curvature tensor

If $X, Y, Z , W$ are vector fields in $M$ and $∇$ is the Levi–Civita connection, then

$$\begin{equation} \begin{matrix} R(X, Y, Z , W ) + R(Y, Z , X, W ) + R(Z , X, Y, W ) = 0,\\ R(X, Y, Z , W ) = −R(Y, X, Z , W ),\\ R(X, Y, Z , W ) = −R(X, Y, W, Z ),\\ R(X, Y, Z , W ) = R(Z , W, X, Y ). \end{matrix} \tag{24.2} \end{equation}$$

Since the Levi-Civita connections are torsion-free, the first Bianchi identity (23.4) reduces to $R(X, Y)Z + R(Z, X)Y + R(Y, Z)X=0$ and implies the first relation.

The antisymmetry $R(X, Y) = −R(Y, X)$ implies the second equality.

The third equality is equivalent to $R(X, Y, Z , Z ) = 0.$ Indeed, if it holds then clearly $R(X, Y, Z , Z ) = 0.$ Conversely, if $R(X, Y, Z , Z ) = 0$, we have

$$R(X, Y, Z + W, Z + W ) = 0\\ ⇔R(X, Y, Z , Z ) + R(X, Y, Z, W ) +R(X, Y, W, Z ) + R(X, Y, W, W )=0\\ ⇔ R(X, Y, Z , W ) + R(X, Y, W, Z ) = 0.$$

To prove $R(X, Y, Z , Z ) = 0$, let us note that since the Levi-Civita connection is compatible with the metric, we have

$$g(∇_X ∇_Y Z , Z ))=X ( g(∇_Y Z , Z ))-g( ∇_Y Z , ∇_X Z),\\ g(∇_Y ∇_X Z , Z ))=Y ( g(∇_X Z , Z ))-g( ∇_X Z , ∇_Y Z).$$

It is also clear, that for connections compatible with the metric, taking $V=[X,Y]$ we obtain

$$V(g(Z,Z))=2g(\nabla_V Z,Z) \Longrightarrow [X,Y](g(Z , Z))=2 g(∇_{[X,Y ]} Z , Z).$$

Therefore $$R(X, Y, Z , Z ) = g(∇_X ∇_Y Z , Z) − g(∇_Y ∇_X Z , Z) − g(∇_{[X,Y ]} Z , Z)\\ = X(g( ∇_Y Z , Z)) − g(∇_Y Z , ∇_X Z)\\ − Y(g( ∇_X Z , Z))+ g(∇_X Z , ∇_Y Z)\\ − \frac{1}{2}[X, Y ](g( Z , Z))$$ But we have also $$g(\nabla_YZ,Z)=Y(g(Z,Z))-g(Z,\nabla_YZ),\\ g(\nabla_XZ,Z)=X(g(Z,Z))-g(Z,\nabla_XZ),$$

and consequently $$g(\nabla_YZ,Z)=\frac{1}{2}Y(g(Z,Z)),\\ g(\nabla_XZ,Z)=\frac{1}{2}X(g(Z,Z)),$$

therefore

$$R(X, Y, Z , Z )\\ =\frac{1}{2}X(Y( g(Z , Z ))-\frac{1}{2}Y(X( g(Z , Z ))-\frac{1}{2}[X,Y]( g(Z , Z ))\\ =\frac{1}{2}[X,Y]( g(Z , Z ))-\frac{1}{2}[X,Y]( g(Z , Z ))=0.$$

In the proof of the fourth relation we use the first one to obtain

$$R(X, Y, Z , W )+R(Y, Z , X, W )+R(Z , X, Y, W )=0,\\ R(Y, Z, W, X) +R(Z , W, Y, X )+R(W, Y, Z , X )=0,\\ R(Z , W, X, Y )+R(W, X, Z , Y )+R(X, Z , W, Y )=0,\\ R(W, X, Y, Z )+R(X, Y, W, Z )+R(Y, W, X, Z )=0.$$

Adding side by side and using the third equality we obtain

$$R(Z , X, Y, W ) + R(W, Y, Z , X ) + R(X, Z , W, Y ) + R(Y, W, X, Z ) = 0.$$

Using the second and third relation, we have

$$2R(Z , X, Y, W ) − 2R(Y, W, Z , X ) = 0.$$

The component-wise version of (24.2) reads as follows:

$$\begin{equation} \begin{matrix} R_{i jkl} + R_{ikl j} + R_{il jk} = 0,\\ R_{ijkl} = −R_{jikl},\\ R_{ijkl} = −R_{ijlk},\\ R_{ijkl} = R_{klij}. \end{matrix} \tag{24.2'} \end{equation}$$

The case of of 2-dimensional manifolds in $R^3$

If $n=2$, then the indices take values 1 or 2 (or 0,1 if Python is used). The antisymmetry in $i,j$ and $k,l$ means that for nonzero components of of the tensor $R_{ijkl}$, we have $i\not=j$ and $k\not=l$. Furthermore changing the order of $i,j$ or $k,l$ we only change the sign of the corresponding components $R_{ijkl}$. Thus, for manifolds of dimension $n = 2,$ there is only one independent component of the Riemannian curvature tensor, namely, $R_{1212}$. The remaining ones vanish or can be obtained by permutations of indices $(i,j)=(1,2), \ (k,l)=(1,2)$ and an appropriate change of signs.

Suppose we are given a surface in Euclidean 3-space

$$x=x(u,v),\ \ y=y(u,v),\ \ z=z(u,v),\ \ (u,v)\in V\subset R^2,$$

and a non-singular point $P=(x_0 , y_0 , z_0 )=(x(u_0,v_0),y(u_0,v_0),z(u_0,v_0))$ on it. i.e.,

$$\text{rank}\left. \left( \begin{matrix} x_u & y_u & z_u\\ x_v & y_v & z_v \end{matrix} \right) \right|_{(u_0,v_0)}=2.$$

We shall assume that the $z$-axis is perpendicular to the tangent plane to the surface at the point $P,$ in which case the $x$-axis and $y$-axis will be parallel to it. The surface may then be given locally about (i.e. in a neighborhood of) the point $P$ by an equation of the form $z = f(x, y)$, where $z_0 = f(x_0 , y_0 )$ and

$$\frac{\partial f}{\partial x}\Big|_{\substack{x=x_0\\y=y_0}} =\frac{\partial f}{\partial y}\Big|_{\substack{x=x_0\\y=y_0}}=0, \quad \text{i.e.}\quad \text{grad}\, f\,\Big|_{\substack{x=x_0\\y=y_0}}=0.$$

Given a surface $z = f(x, y)$, and a point $P=(x_0 , y_0 , z_0 )$ on it at which grad $f = 0$, we can define the Gaussian curvature $K$ of the surface at $(x_0,y_0,z_0)$ to be the determinant of the matrix

$$K=\det (f_{x^ix^j})\Big|_{\substack{x=x_0\\y=y_0}}= \det \left( \begin{matrix} f_{xx} & f_{xy}\\ f_{yx} & f_{yy} \end{matrix} \right)\Big|_{\substack{x=x_0\\y=y_0}}.$$

The metric on such surface is defined by

$$g_{11}=1+f^2_x,\quad g_{12}=f_xf_y,\quad g_{22}=1+f_y^2.$$

By the assumption $\text{grad}\, f\,\Big|_{\substack{x=x_0\\y=y_0}}=0$ at point $P$, the derivatives of $g_{ij}$ with respect to $x$ and $y$ vanish, so all Levi-Civita connection coefficients $\Gamma^m_{ij}=0$ at $P$.

Recall the formulas defining the Riemannian curvature tensor at $P$ ( here, as usual $(g_{ij})$ is the matrix of metric components and $(g^{ij})$ its inverse).

$$\displaystyle\Gamma_{rs}^q=\frac{1}{2}g^{tq}\Big(\frac{\partial g_{rt}}{\partial x^s} +\frac{\partial g_{ts}}{\partial x^r}- \frac{\partial g_{sr}}{\partial x^t}\Big),\\ R^m_{kij}=\frac{\partial \Gamma^m_{kj}}{\partial x^i} -\frac{\partial \Gamma^m_{ki}}{\partial x^j} +\Gamma^m_{li}\Gamma^l_{kj} -\Gamma^m_{lj}\Gamma^l_{ki},\\ R_{ijkl}=g_{im}R^m_{jkl} =g_{im} \Big(\frac{\partial \Gamma^m_{jl}}{\partial x^k} -\frac{\partial \Gamma^m_{jk}}{\partial x^l}\Big),\\$$

(in the last formula we used the fact that $\Gamma^m_{ki}=0$ at $P=(x_0,y_0,z_0)$). To obtain the derivatives of Christoffel symbols we need their values in a neighborhood of $P$

$$\Gamma^m_{jl}=\frac{1}{2}g^{pm}\Big(\frac{\partial g_{jp}}{\partial x^l} +\frac{\partial g_{pl}}{\partial x^j}- \frac{\partial g_{lj}}{\partial x^p}\Big),$$

so using the fact that derivatives of $g_{ij}$ vanish at $P$ we obtain

$$\frac{\partial \Gamma^m_{jl}}{\partial x^k} =\frac{1}{2}g^{pm}\big(\frac{\partial^2 g_{jp}}{\partial x^k\partial x^l} +\frac{\partial^2 g_{pl}}{\partial x^k\partial x^j}- \frac{\partial^2 g_{lj}}{\partial x^k\partial x^p}\Big).$$

Similarly

$$\Gamma^m_{jk}=\frac{1}{2}g^{pm}\Big(\frac{\partial g_{jp}}{\partial x^k} +\frac{\partial g_{pk}}{\partial x^j}- \frac{\partial g_{kj}}{\partial x^p}\Big),$$

so $$\frac{\partial \Gamma^m_{jk}}{\partial x^l} =\frac{1}{2}g^{pm}\big(\frac{\partial^2 g_{jp}}{\partial x^l\partial x^k} +\frac{\partial^2 g_{pk}}{\partial x^l\partial x^j}- \frac{\partial^2 g_{kj}}{\partial x^l\partial x^p}\Big).$$

Since $g_{im}g^{pm}=\delta_i^p$, then $R_{ijkl}$ is a difference of two expressions:

$$\frac{1}{2}\delta_i^p\big(\frac{\partial^2 g_{jp}}{\partial x^k\partial x^l} +\frac{\partial^2 g_{pl}}{\partial x^k\partial x^j}- \frac{\partial^2 g_{lj}}{\partial x^k\partial x^p}\big) = \frac{1}{2}\big(\frac{\partial^2 g_{ji}}{\partial x^k\partial x^l} +\frac{\partial^2 g_{il}}{\partial x^k\partial x^j}- \frac{\partial^2 g_{lj}}{\partial x^k\partial x^i}\big)$$

and

$$\frac{1}{2}\delta_i^p\big(\frac{\partial^2 g_{jp}}{\partial x^l\partial x^k} +\frac{\partial^2 g_{pk}}{\partial x^l\partial x^j}- \frac{\partial^2 g_{kj}}{\partial x^l\partial x^p}\big) = \frac{1}{2}\big(\frac{\partial^2 g_{ji}}{\partial x^l\partial x^k} +\frac{\partial^2 g_{ik}}{\partial x^l\partial x^j}- \frac{\partial^2 g_{kj}}{\partial x^l\partial x^i}\big).$$

Thus

$$R_{ijkl}=\frac{1}{2}\Big( \frac{\partial^2 g_{il}}{\partial x^k\partial x^j}+ \frac{\partial^2 g_{kj}}{\partial x^l\partial x^i}- \frac{\partial^2 g_{lj}}{\partial x^k\partial x^i}- \frac{\partial^2 g_{ik}}{\partial x^l\partial x^j}\Big).$$

For $(i,j,k,l)=(1,2,1,2)$ we obtain $$R_{1212}=\frac{1}{2}\Big( \frac{\partial^2 g_{12}}{\partial x^1\partial x^2}+ \frac{\partial^2 g_{12}}{\partial x^2\partial x^1}- \frac{\partial^2 g_{22}}{\partial x^1\partial x^1}- \frac{\partial^2 g_{11}}{\partial x^2\partial x^2}\Big)\\ =\frac{1}{2}\Big(\frac{\partial^2 g_{12}}{\partial x\partial y}+ \frac{\partial^2 g_{12}}{\partial y\partial x}- \frac{\partial^2 g_{22}}{\partial x^2}- \frac{\partial^2 g_{11}}{\partial y^2}\Big) .$$

Since at $P$ $$(g_{11})_{yy}=2(f_{xy})^2,\quad (g_{22})_{xx}=2(f_{xy})^2,\quad (g_{12})_{xy}=f_{xx}f_{yy}+(f_{xy})^2,$$ we have at $P$ $$R_{1212}=\frac{1}{2}\big(2(f_{xx}f_{yy}+(f_{xy})^2) -2(f_{xy})^2 -2(f_{xy})^2\big)=f_{xx}f_{yy}-(f_{xy})^2=\det \left( \begin{matrix} f_{xx} & f_{xy}\\ f_{yx} & f_{yy} \end{matrix} \right)=K.$$

Ricci tensor and scalar curvature

Ricci tensor can be defined by

$$\begin{equation} Ric(u,v)=R(e^i,u,e_i,v), \tag{24.3} \end{equation}$$

for any vector fields $u,v$, where $e_i$ is any vector frame and $e^i$ the corresponding coframe.

In a Riemannian manifold with the metric $g$ the components of Ricci tensor are defined by

$$Ric_{ij}=R^k_{ikj}=g^{kl}R_{likj}=g^{lk}R_{kjli},$$

and the scalar curvature by

$$\begin{equation} R=g^{ij}Ric_{ij}. \tag{24.4} \end{equation}$$

In the case of two-dimensional manifold we have $R=2K$. In fact

$$R=g^{ij}Ric_{ij}=g^{ij}g^{lk}R_{kjli}.$$

From the symmetry properties of $R_{ijkl}$ it follows that in the obtained sum of $2^4$ elements only 4 are non-zero:

$$R=g^{22}g^{11}R_{1212}+g^{12}g^{21}R_{1221}+ g^{21}g^{12}R_{2112}+g^{11}g^{22}R_{2121}\\ =g^{22}g^{11}R_{1212}-g^{12}g^{21}R_{1212}-g^{21}g^{12}R_{1212}+g^{11}g^{22}R_{1212}\\ =2(g^{11}g^{22}-(g^{12})^2)R_{1212}=2\det (g^{ij})R_{1212}=\frac{2}{\det (g_{ij})}R_{1212}.$$

Remark in the last calculations the condition $\text{grad}\, f\,\Big|_{\substack{x=x_0\\y=y_0}}=0,$ was not used.

If as previously, the coordinates at $P$ are such that $\text{grad}\, f\,\Big|_{\substack{x=x_0\\y=y_0}}=0,$ we have $\det (g_{ij})\Big|_{\substack{x=x_0\\y=y_0}}=1$ and $R=2K$. Since $R$ and $K$ are scalars, and consequently, are coordinate-independent, we conclude that

$$\begin{equation} R = 2K \tag{24.5} \end{equation}$$

at every (non-singular) point of the surface.

Remark. For non-singular surfaces represented locally in the form $z=f(x,y)$ (without the condition $\text{grad}\, f\,\Big|_{\substack{x=x_0\\y=y_0}}=0$) the Gauss curvature is defined by $$K=\frac{f_{xx}f_{yy}-(f_{xy})^2}{(1+(f_x)^2+(f_y)^2)^2}.$$

Remark. The calculations above give us the formula $$R_{1212}=\frac{R}{2}\det(g_{ij}).$$

It can be generalized to all components of the tensor $R_{ijkl}$ as follows:

$$R_{ijkl}=\frac{R}{2}(g_{ik}g_{jl}-g_{il}g_{jk}).$$

In fact, an explicit calculations using the symmetry properties of $R_{ijkl}$ show that

$$\begin{darray}{rcl} (i,j,k,l)=(1,2,1,2)\quad \Rightarrow & \quad \frac{R}{2}(g_{11}g_{22}-g_{12}g_{21})=\frac{R}{2}\det(g_{ij})=R_{1212},\\ (i,j,k,l)=(2,1,1,2)\quad \Rightarrow & \quad \frac{R}{2}(g_{21}g_{12}-g_{22}g_{11})=-\frac{R}{2}\det(g_{ij})=R_{2112},\\ (i,j,k,l)=(1,2,2,1)\quad \Rightarrow & \quad \frac{R}{2}(g_{12}g_{21}-g_{11}g_{22})=-\frac{R}{2}\det(g_{ij})=R_{1221},\\ (i,j,k,l)=(2,1,2,1)\quad \Rightarrow & \quad \frac{R}{2}(g_{22}g_{11}-g_{21}g_{12})=\frac{R}{2}\det(g_{ij})=R_{2121},\\ \end{darray}$$

and for the remaining sequences of indices, both sides are zero.

Example 24.1

In $R^2$ with $g=dx\otimes dx+dy\otimes dy$, we have $R^i_{mkl}=0$ and $K=\frac{R}{2}=0.$

E.<x,y>=EuclideanSpace()      # Euclidean space E^2
print(E.metric().riemann())   # curvature tensor of (1,3)-type
E.metric().riemann().disp()   # show the curvature tensor

Tensor field Riem(g) of type (1,3) on the Euclidean plane E^2

Riem(g) = 0

The method ricci gives the Ricci tensor:

print(E.metric().ricci())    # Ricci curvature tensor
E.metric().ricci().disp()    # show the result

Field of symmetric bilinear forms Ric(g) on the Euclidean plane E^2

Ric(g) = 0

The Ricci scalar curvature can be obtained using ricci_scalar method:

E.metric().ricci_scalar().expr()  # Ricci scalar curvature

0

Example 24.2

The same can be done without EuclideanSpace command.

%display latex
M=Manifold(2,name='R^2',start_index=1) # manifold R2
c_xy.<x,y>=M.chart()        # chart on R^2
g = M.metric('g');          # metric on R^2
g[:]=[[1,0],[0,1]]          # nonzero components
Riem=g.riemann()            # curvature (1,3)-type tensor 
print(Riem)                 # show information on curv. tensor
Riem.disp()                 # show the curvature tensor

Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold R^2

$\displaystyle \mathrm{Riem}\left(g\right) = 0$

print(g.ricci())            # Ricci curvature tensor information
g.ricci().disp()            # show  Ricci tensor

Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold R^2

$\displaystyle \mathrm{Ric}\left(g\right) = 0$

The Ricci scalar curvature:

g.ricci_scalar().expr()     # Ricci scalar curvature 

$\displaystyle 0$

Example 24.3

For the metric on the sphere $S^2\subset R^3$, we have $𝑔=d𝜃⊗d𝜃+sin(𝜃)^2d𝜙⊗d𝜙$, $K=R/2=1.$

%display latex
M = Manifold(3, 'R^3')              # R^3 ambient space for S^2
c_xyz.<x,y,z> = M.chart()           # chart on R^3
N = Manifold(2, 'N',start_index=1)  # sphere S^2
c_sph.<theta,phi>=N.chart()         # chart on S^2
psi = N.diff_map(M, (sin(theta)*cos(phi), 
                     sin(theta)*sin(phi),
                     cos(theta)),
   name='psi',latex_name=r'\psi')   # embedding S^2 -> R^3
g0=M.metric('g0')                   # metric g0 on R^3
g0[0,0],g0[1,1],g0[2,2]=1,1,1       # components of g0
g=N.metric('g')                     # metric on S^2
g.set( psi.pullback(g0) )           # g is the pullback of g0
g.disp()                            # show g

$\displaystyle g = \mathrm{d} \theta\otimes \mathrm{d} \theta + \sin\left(\theta\right)^{2} \mathrm{d} \phi\otimes \mathrm{d} \phi$

Riem=g.riemann()                    # curvature (1,3)-tensor
Riem.display_comp(coordinate_labels=False)  # show components

$\displaystyle \begin{array}{lcl} \mathrm{Riem}\left(g\right)_{\phantom{\, 1}\,2\,1\,2}^{\,1\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}} & = & \sin\left(\theta\right)^{2} \\ \mathrm{Riem}\left(g\right)_{\phantom{\, 1}\,2\,2\,1}^{\,1\phantom{\, 2}\phantom{\, 2}\phantom{\, 1}} & = & -\sin\left(\theta\right)^{2} \\ \mathrm{Riem}\left(g\right)_{\phantom{\, 2}\,1\,1\,2}^{\,2\phantom{\, 1}\phantom{\, 1}\phantom{\, 2}} & = & -1 \\ \mathrm{Riem}\left(g\right)_{\phantom{\, 2}\,1\,2\,1}^{\,2\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}} & = & 1 \end{array}$

print(g.ricci())                   # Ricci curvature tensor
g.ricci().disp()                   # show Ricci tensor

Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold N

$\displaystyle \mathrm{Ric}\left(g\right) = \mathrm{d} \theta\otimes \mathrm{d} \theta + \sin\left(\theta\right)^{2} \mathrm{d} \phi\otimes \mathrm{d} \phi$

Let us show how to apply (24.3).

First we have to define the frame and coframe fields:

fr=c_sph.frame()[:]                # frame
cfr=c_sph.coframe()[:]             # coframe

next we define the matrix of coefficient of the Ricci tensor field according to (24.3).

Ric_coeff=[[sum([Riem(cfr[i],fr[k],fr[i],fr[l]).expr() 
                 for i in range(2)])
 for k in range(2)] for l in range(2)]  # cf. (24.3)
Ric_coeff[:]                       # Ricci ten. components

$\displaystyle \left[\left[1, 0\right], \left[0, \sin\left(\theta\right)^{2}\right]\right]$

Finally we define the Ricci tensor:

Ric=N.tensor_field(0,2,'Ric')     # Ricci curvature tensor
Ric[:]=Ric_coeff                  # define all components
Ric.disp()                        # show Ricci tensor

$\displaystyle Ric = \mathrm{d} \theta\otimes \mathrm{d} \theta + \sin\left(\theta\right)^{2} \mathrm{d} \phi\otimes \mathrm{d} \phi$

Ricci scalar curvature:

g.ricci_scalar().expr()           # Ricci scalar curvature

$\displaystyle 2$

Formula (24.4) can be used as follows:

g.inverse().contract(Ric).trace().expr()

$\displaystyle 2$

According to (24.5), the Gaussian curvature is 1.

Example 24.4

Consider the Poincare half-plane $y>0$ with the metric defined by $g_{11}=\frac{1}{y^2},\ \ g_{22}=\frac{1}{y^2}, \$ and remaining components equal to zero.

M = Manifold(2, 'M', start_index=1)  # Poincare half-plane
X.<x,y> = M.chart(coord_restrictions=lambda x,y: y>0)
g = M.metric('g')                    # metric on M
g[1,1], g[2,2] = 1/y^2, 1/y^2        # nonzero components of g

Compute (1,3) type curvature tensor:

Riem=g.riemann()                     # (1,3) type curvature tensor
Riem.display_comp(coordinate_labels=False)  # show nonzero comp.

$\displaystyle \begin{array}{lcl} \mathrm{Riem}\left(g\right)_{\phantom{\, 1}\,2\,1\,2}^{\,1\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}} & = & -\frac{1}{y^{2}} \\ \mathrm{Riem}\left(g\right)_{\phantom{\, 1}\,2\,2\,1}^{\,1\phantom{\, 2}\phantom{\, 2}\phantom{\, 1}} & = & \frac{1}{y^{2}} \\ \mathrm{Riem}\left(g\right)_{\phantom{\, 2}\,1\,1\,2}^{\,2\phantom{\, 1}\phantom{\, 1}\phantom{\, 2}} & = & \frac{1}{y^{2}} \\ \mathrm{Riem}\left(g\right)_{\phantom{\, 2}\,1\,2\,1}^{\,2\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}} & = & -\frac{1}{y^{2}} \end{array}$

and next the Riemannian curvature tensor of type (0,4), using (24.1):

R=M.tensor_field(0,4,'R')         # R is (0,4)-type tensor field
R[:]=g.contract(Riem)[:]                # apply (24.1)
R.display_comp(coordinate_labels=False) # show (0,4) type ten. R

$\displaystyle \begin{array}{lcl} R_{\,1\,2\,1\,2}^{\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}} & = & -\frac{1}{y^{4}} \\ R_{\,1\,2\,2\,1}^{\phantom{\, 1}\phantom{\, 2}\phantom{\, 2}\phantom{\, 1}} & = & \frac{1}{y^{4}} \\ R_{\,2\,1\,1\,2}^{\phantom{\, 2}\phantom{\, 1}\phantom{\, 1}\phantom{\, 2}} & = & \frac{1}{y^{4}} \\ R_{\,2\,1\,2\,1}^{\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}} & = & -\frac{1}{y^{4}} \end{array}$

Now compute the Ricci curvature tensor:

g.ricci().disp()                      # Ricci curv. tensor

$\displaystyle \mathrm{Ric}\left(g\right) = -\frac{1}{y^{2}} \mathrm{d} x\otimes \mathrm{d} x -\frac{1}{y^{2}} \mathrm{d} y\otimes \mathrm{d} y$

and Ricci curvature scalar:

g.ricci_scalar().expr()               # Ricci scalar curv.

$\displaystyle -2$

Example 24.5

Poincare disk is the set $\ \ \{(x,y)\in R^2:x^2+y^2<1\}\ \$ with the metric defined by $g_{11}=\frac{4}{(1-x^2-y^2)^2},\ \ g_{22}=\frac{4}{(1-x^2-y^2)^2}\ \$ and remaining components equal to zero.

M = Manifold(2, 'M', start_index=1)  # Poincare disc
                # Cartesian coordinates on the Poincaré disk:
X.<x,y> = M.chart('x:(-1,1) y:(-1,1)', 
    coord_restrictions=lambda x,y: x^2+y^2<1)
g = M.metric('g')                    #  metric on M
                                     # components of g
g[1,1], g[2,2] = 4/(1-x^2-y^2)^2,4/(1-x^2-y^2)^2

Compute (1,3)-type curvature tensor. Since g.riemann() gives an unsimplified result, we simplify the components one by one.

Riem = g.riemann();           # curvature (1,3) type tensor
                              # factor components
Riem2=[[[[Riem[a,b,c,d].factor() for a in [1,2]] 
         for b in [1,2]] for c in [1,2]] for d in[1,2]];
R=M.tensor_field(1,3,'R')     #  tensor field of type (1,3)
R[X.frame(),:]=Riem2          # simplified components
R.display_comp(coordinate_labels=False)  # show R

$\displaystyle \begin{array}{lcl} R_{\phantom{\, 1}\,2\,1\,2}^{\,1\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}} & = & -\frac{4}{{\left(x^{2} + y^{2} - 1\right)}^{2}} \\ R_{\phantom{\, 1}\,2\,2\,1}^{\,1\phantom{\, 2}\phantom{\, 2}\phantom{\, 1}} & = & \frac{4}{{\left(x^{2} + y^{2} - 1\right)}^{2}} \\ R_{\phantom{\, 2}\,1\,1\,2}^{\,2\phantom{\, 1}\phantom{\, 1}\phantom{\, 2}} & = & \frac{4}{{\left(x^{2} + y^{2} - 1\right)}^{2}} \\ R_{\phantom{\, 2}\,1\,2\,1}^{\,2\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}} & = & -\frac{4}{{\left(x^{2} + y^{2} - 1\right)}^{2}} \end{array}$

Ricci curvature tensor:

Ric=g.ricci()                # Ricci curvature tensor
Ric.apply_map(factor)        # factor components
Ric.disp()                   # show Ricci curv.tensor  

$\displaystyle \mathrm{Ric}\left(g\right) = -\frac{4}{{\left(x^{2} + y^{2} - 1\right)}^{2}} \mathrm{d} x\otimes \mathrm{d} x -\frac{4}{{\left(x^{2} + y^{2} - 1\right)}^{2}} \mathrm{d} y\otimes \mathrm{d} y$

Ricci curvature scalar:

g.ricci_scalar().expr()     # Ricci scalar curvature 

$\displaystyle -2$

Example 24.6

Consider $R^2$ with the matrix of metric components:

matrix([[1+4*x^2,4*x*y],[4*x*y,1+4*y^2]])

$\displaystyle \left(\begin{array}{rr} 4 \, x^{2} + 1 & 4 \, x y \\ 4 \, x y & 4 \, y^{2} + 1 \end{array}\right)$

N=Manifold(2,name='R^2',start_index=1) # Manifold R^2
X.<x,y>=N.chart()                      # chart on R^2
g = N.metric('g');                     # metric g on R^2
g[:]=[[1+4*x^2,4*x*y],[4*x*y,1+4*y^2]] # components of g
Riem = g.riemann();                    # (1,3) type curv. tens.
R=N.tensor_field(0,4,'R')              # (0,4) type curv. tens.

Compute Riemannian (0,4)-type tensor.

R[:]=g.contract(Riem)[:]                   # use (24.1)
R.display_comp(coordinate_labels=False)    # show (0,4) t. R

$\displaystyle \begin{array}{lcl} R_{\,1\,2\,1\,2}^{\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}} & = & \frac{4}{4 \, x^{2} + 4 \, y^{2} + 1} \\ R_{\,1\,2\,2\,1}^{\phantom{\, 1}\phantom{\, 2}\phantom{\, 2}\phantom{\, 1}} & = & -\frac{4}{4 \, x^{2} + 4 \, y^{2} + 1} \\ R_{\,2\,1\,1\,2}^{\phantom{\, 2}\phantom{\, 1}\phantom{\, 1}\phantom{\, 2}} & = & -\frac{4}{4 \, x^{2} + 4 \, y^{2} + 1} \\ R_{\,2\,1\,2\,1}^{\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}} & = & \frac{4}{4 \, x^{2} + 4 \, y^{2} + 1} \end{array}$

Ricci curvature tensor:

Ric=g.ricci()                          # Ricci curvature tensor
Ric.apply_map(factor)                  # factor components
Ric.disp()                             # show the result

$\displaystyle \mathrm{Ric}\left(g\right) = \frac{4 \, {\left(4 \, x^{2} + 1\right)}}{{\left(4 \, x^{2} + 4 \, y^{2} + 1\right)}^{2}} \mathrm{d} x\otimes \mathrm{d} x + \frac{16 \, x y}{{\left(4 \, x^{2} + 4 \, y^{2} + 1\right)}^{2}} \mathrm{d} x\otimes \mathrm{d} y + \frac{16 \, x y}{{\left(4 \, x^{2} + 4 \, y^{2} + 1\right)}^{2}} \mathrm{d} y\otimes \mathrm{d} x + \frac{4 \, {\left(4 \, y^{2} + 1\right)}}{{\left(4 \, x^{2} + 4 \, y^{2} + 1\right)}^{2}} \mathrm{d} y\otimes \mathrm{d} y$

and Ricci scalar curvature:

factor(g.ricci_scalar().expr()/2)      # Ricci curvature scalar

$\displaystyle \frac{4}{{\left(4 \, x^{2} + 4 \, y^{2} + 1\right)}^{2}}$

Example 24.7

Consider the embedding of the hyperboloid into $R^3$ defined by $\ \ \psi(u,v)= (u, v,u^2-v^2)$. The metric on the hyperboloid can be defined as the pullback $\psi^*g_0\ \$ under $\psi$ where $g_0$ is the standard metric on $R^3$.

%display latex
M = Manifold(3, 'R3')                  # manifold R^3
c_xyz.<x,y,z> = M.chart()              # Cart. coord. on R^3
N=Manifold(2,name='R2',start_index=1)  # hyperboloid N
c_uv.<u,v>=N.chart()                   # coordinates on N
g0 = M.metric('g0');                   # standard metr, of R^3
g0[:]=[[1,0,0],[0,1,0],[0,0,1]];       # components of g0
psi = N.diff_map(M, (u, v,u^2-v^2) ,name='psi')  # embedding
g=N.metric('g')                        # metric on N
g.set(psi.pullback(g0))                # g is pullback of g0
g.disp()                               # show g

$\displaystyle g = \left( 4 \, u^{2} + 1 \right) \mathrm{d} u\otimes \mathrm{d} u -4 \, u v \mathrm{d} u\otimes \mathrm{d} v -4 \, u v \mathrm{d} v\otimes \mathrm{d} u + \left( 4 \, v^{2} + 1 \right) \mathrm{d} v\otimes \mathrm{d} v$

Compute the Riemannian curvature tensor of type (0,4).

Riem = g.riemann();                    # curv. (1,3) tensor
R=N.tensor_field(0,4,'R')              # tensor field (0,4)-type
R[:]=g.contract(Riem)[:]               # use (24.1)
R.display_comp(coordinate_labels=False) # show (0,4) type ten. R

$\displaystyle \begin{array}{lcl} R_{\,1\,2\,1\,2}^{\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}} & = & -\frac{4}{4 \, u^{2} + 4 \, v^{2} + 1} \\ R_{\,1\,2\,2\,1}^{\phantom{\, 1}\phantom{\, 2}\phantom{\, 2}\phantom{\, 1}} & = & \frac{4}{4 \, u^{2} + 4 \, v^{2} + 1} \\ R_{\,2\,1\,1\,2}^{\phantom{\, 2}\phantom{\, 1}\phantom{\, 1}\phantom{\, 2}} & = & \frac{4}{4 \, u^{2} + 4 \, v^{2} + 1} \\ R_{\,2\,1\,2\,1}^{\phantom{\, 2}\phantom{\, 1}\phantom{\, 2}\phantom{\, 1}} & = & -\frac{4}{4 \, u^{2} + 4 \, v^{2} + 1} \end{array}$