DifferentialGeometry/Tensor/SectionalCurvature - Maple Help

Tensor[SectionalCurvature] - calculate the sectional curvature for a metric 

Calling Sequences

SectionalCurvature(g, R, X, Y)

Parameters

g    - a metric tensor on the tangent bundle of a manifold

R    - the curvature tensor of the metric $g,$calculated from the Christoffel symbol of $g$

X, Y - a pair of vectors

Description

 • Let $M$ be an $n$-dimensional manifold with metric $g$. The sectional curvature of the metric $g$ at a point $p\in M$ is the Gaussian curvature $K$ (at $p$) of the geodesic surface whose tangent space at is spanned by vectors ${X}_{}$ and ${Y}_{}$. If $R$ is the covariant form of the curvature tensor (that is, $R$ is a tensor of type $\left(\genfrac{}{}{0}{}{0}{4}\right)$), then

.

 • If $K$ is independent of the choice of the vectors $X$ and $Y$ then $K=\frac{S}{n\left(n-1\right)}$, where $S$ is the Ricci scalar of $g$.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SectionalCurvature(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:- SectionalCurvature(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

First create a 2 dimensional manifold $\mathrm{M1}$ and define a metric $\mathrm{g1}$ on $\mathrm{M1}$.

 M4 > $\mathrm{DGsetup}\left(\left[x,y\right],\mathrm{M1}\right)$
 ${\mathrm{frame name: M1}}$ (2.1)
 M1 > $\mathrm{g1}≔\mathrm{evalDG}\left(\frac{1\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}\right)}{{y}^{2}}\right)$
 ${\mathrm{g1}}{:=}\frac{{\mathrm{dx}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dy}}}{{{y}}^{{2}}}$ (2.2)

Compute the sectional curvature determined by the coordinate basis vectors ${\partial }_{x}$ and ${\partial }_{y}$ .

 M1 > $\mathrm{R1}≔\mathrm{CurvatureTensor}\left(\mathrm{g1}\right)$
 ${\mathrm{R1}}{:=}{-}\frac{{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{2}}}{+}\frac{{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{+}\frac{{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{2}}}{-}\frac{{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{2}}}$ (2.3)
 M1 > $K≔\mathrm{SectionalCurvature}\left(\mathrm{g1},\mathrm{R1},\mathrm{D_x},\mathrm{D_y}\right)$
 ${K}{:=}{-}{1}$ (2.4)

For 2-dimensional manifolds the sectional curvature coincides with the Gaussian curvature ${R}_{1212}}{\mathrm{det}\left(g\right)}$. Let us check this formula.

 M1 > $R≔\mathrm{RaiseLowerIndices}\left(\mathrm{g1},\mathrm{R1},\left[1\right]\right)$
 ${R}{:=}{-}\frac{{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{4}}}{+}\frac{{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{4}}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{4}}}{-}\frac{{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{4}}}$ (2.5)
 M1 > $\mathrm{R1212}≔{\mathrm{Tools}:-\mathrm{DGinfo}\left(R,"CoefficientList",\left[\left[1,2,1,2\right]\right]\right)}_{1}$
 ${\mathrm{R1212}}{:=}{-}\frac{{1}}{{{y}}^{{4}}}$ (2.6)
 M1 > $\frac{\mathrm{R1212}}{\mathrm{MetricDensity}\left(\mathrm{g1},2\right)}$
 ${-}{1}$ (2.7)

Example 2.

First create a 3 dimensional manifold $\mathrm{M2}$ and define a metric $\mathrm{g2}$ on $\mathrm{M2}$.

 M1 > $\mathrm{DGsetup}\left(\left[x,y,z\right],\mathrm{M2}\right)$
 ${\mathrm{frame name: M2}}$ (2.8)
 M2 > $\mathrm{g2}≔\mathrm{evalDG}\left(\frac{{a}^{2}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)}{{\left({k}^{2}+{x}^{2}+{y}^{2}+{z}^{2}\right)}^{2}}\right)$
 ${\mathrm{g2}}{:=}\frac{{{a}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}{+}\frac{{{a}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}{+}\frac{{{a}}^{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}$ (2.9)

Define a pair of vectors which span a generic tangent plane.

 M2 > $X≔\mathrm{evalDG}\left(\mathrm{D_x}+r\mathrm{D_y}+s\mathrm{D_y}\right)$
 ${X}{:=}{\mathrm{D_x}}{+}\left({s}{+}{r}\right){}{\mathrm{D_y}}$ (2.10)
 M2 > $Y≔\mathrm{evalDG}\left(\mathrm{D_y}+t\mathrm{D_z}\right)$
 ${Y}{:=}{\mathrm{D_y}}{+}{t}{}{\mathrm{D_z}}$ (2.11)

Calculate the curvature and sectional curvature. Note that the sectional curvature is independent of the parameters $r,s,t$ appearing in the vector fields $X$ and $Y$.

 M2 > $\mathrm{R2}≔\mathrm{CurvatureTensor}\left(\mathrm{g2}\right):$
 M2 > $\mathrm{K2}≔\mathrm{SectionalCurvature}\left(\mathrm{g2},\mathrm{R2},X,Y\right)$
 ${\mathrm{K2}}{:=}\frac{{4}{}{{k}}^{{2}}}{{{a}}^{{2}}}$ (2.12)

Since the metric $\mathrm{g2}$ has constant sectional curvature and the dimension of $\mathrm{M2}$ is $3$, the sectional curvature is 1/6 the Ricci scalar.

 M2 > $\mathrm{S2}≔\mathrm{RicciScalar}\left(\mathrm{g2},\mathrm{R2}\right)$
 ${\mathrm{S2}}{:=}\frac{{24}{}{{k}}^{{2}}}{{{a}}^{{2}}}$ (2.13)

Example 3.

We re-work the previous example in an orthonormal frame.

 M2 > $f≔\frac{a}{{k}^{2}+{x}^{2}+{y}^{2}+{z}^{2}}$
 ${f}{:=}\frac{{a}}{{{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}}$ (2.14)
 M2 > $\mathrm{FR}≔\mathrm{FrameData}\left(\left[f\mathrm{dx},f\mathrm{dy},f\mathrm{dz}\right],\mathrm{M3}\right):$
 M2 > $\mathrm{DGsetup}\left(\mathrm{FR}\right)$
 ${\mathrm{frame name: M3}}$ (2.15)
 M3 > $\mathrm{g3}≔\mathrm{evalDG}\left(\mathrm{Θ1}&t\mathrm{Θ1}+\mathrm{Θ2}&t\mathrm{Θ2}+\mathrm{Θ3}&t\mathrm{Θ3}\right)$
 ${\mathrm{g3}}{:=}{\mathrm{Θ1}}{}{\mathrm{Θ1}}{+}{\mathrm{Θ2}}{}{\mathrm{Θ2}}{+}{\mathrm{Θ3}}{}{\mathrm{Θ3}}$ (2.16)

Calculate the sectional curvature.

 M3 > $\mathrm{R3}≔\mathrm{CurvatureTensor}\left(\mathrm{g3}\right):$
 M3 > $\mathrm{K3}≔\mathrm{SectionalCurvature}\left(\mathrm{g3},\mathrm{R3},\mathrm{E1}+r\mathrm{E2}+t\mathrm{E3},\mathrm{E2}+t\mathrm{E3}\right)$
 ${\mathrm{K3}}{:=}\frac{{4}{}{{k}}^{{2}}}{{{a}}^{{2}}}$ (2.17)

Example 4.

First create a 3 dimensional manifold $\mathrm{M4}$ and define a metric $\mathrm{g4}$ on $\mathrm{M4}$.

 M3 > $\mathrm{DGsetup}\left(\left[x,y,z\right],\mathrm{M4}\right)$
 ${\mathrm{frame name: M4}}$ (2.18)
 M4 > $\mathrm{g4}≔\mathrm{evalDG}\left(y\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)$
 ${\mathrm{g4}}{:=}{y}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.19)

Define a pair of vectors which span a generic tangent plane.

 M4 > $X≔\mathrm{evalDG}\left(\mathrm{D_x}+r\mathrm{D_y}+s\mathrm{D_z}\right)$
 ${X}{:=}{\mathrm{D_x}}{+}{r}{}{\mathrm{D_y}}{+}{s}{}{\mathrm{D_z}}$ (2.20)
 M4 > $Y≔\mathrm{evalDG}\left(\mathrm{D_y}+t\mathrm{D_z}\right)$
 ${Y}{:=}{\mathrm{D_y}}{+}{t}{}{\mathrm{D_z}}$ (2.21)

Calculate the curvature and sectional curvature. In this example, the sectional curvature is dependent on the parameters $r,s,t$ appearing in the vector fields $X$ and $Y$.

 M4 > $\mathrm{R4}≔\mathrm{CurvatureTensor}\left(\mathrm{g4}\right):$
 M4 > $\mathrm{K4}≔\mathrm{SectionalCurvature}\left(\mathrm{g4},\mathrm{R4},X,Y\right)$
 ${\mathrm{K4}}{:=}\frac{{1}}{{4}{}{y}{}\left({{s}}^{{2}}{+}{{t}}^{{2}}{}{{r}}^{{2}}{+}{y}{}{{t}}^{{2}}{+}{y}{-}{2}{}{t}{}{s}{}{r}\right)}$ (2.22)