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Tensor[RicciTensor] - calculate the Ricci tensor of a linear connection on the tangent bundle

Calling Sequences

RicciTensor(g)

RicciTensor(R)

Parameters

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

R    - the curvature tensor of a connection on the tangent bundle of a manifold

Description

 • Let $C$ be a connection on the tangent bundle of a manifold $M$ with a curvature tensor $C$. The Ricci tensor $R$ is the contraction of $C$ over the 1st and 3rd indices. In terms of index notation,  .
 • With the first calling sequence, the Ricci tensor for the Christoffel connection of the metric $g$ is computed. With the second calling sequence, the Ricci tensor is computed directly from the given curvature tensor.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RicciTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-RicciTensor.

Examples

 > with(DifferentialGeometry): with(Tensor):

Example 1.

First create a 3-dimensional manifold $M$ and define a connection on the tangent space of $M$.

 > DGsetup([x, y, z], M);
 ${\mathrm{frame name: M}}$ (2.1)
 M > C1 := Connection(x^2*D_x &t dx &t dy - y^2*D_x &t dy &t dy + y*z* D_x &t dz &t dy);
 ${\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{2}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{1}{,}{2}{,}{2}\right]{,}{-}{{y}}^{{2}}\right]{,}\left[\left[{1}{,}{3}{,}{2}\right]{,}{y}{}{z}\right]\right]\right]\right)$ (2.2)

Calculate the curvature tensor.

 M > R1 := CurvatureTensor(C1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{2}\right]{,}{2}{}{x}\right]{,}\left[\left[{1}{,}{1}{,}{2}{,}{1}\right]{,}{-}{2}{}{x}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{3}\right]{,}{-}{y}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{2}\right]{,}{y}\right]\right]\right]\right)$ (2.3)

Calculate the Ricci tensor. Note that in general the Ricci tensor is not symmetric.

 M > Ric1 := RicciTensor(R1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{2}{}{x}\right]\right]\right]\right)$ (2.4)

Example 2.

Define a frame on $M$ and use this frame to specify a connection on the tangent space of $M$.

 M > DGsetup([x, y, z], M);
 ${\mathrm{frame name: M}}$ (2.5)
 M > FR := FrameData([x^2/y*dx, z/x*dy, x*y*dz], M1):
 M > DGsetup(FR);
 ${\mathrm{frame name: M1}}$ (2.6)
 M1 > C2 := Connection(E2 &t Theta1 &t Theta2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{\mathrm{M1}}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (2.7)

Calculate the curvature tensor.

 M1 > R2 := CurvatureTensor(C2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{M1}}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-}\frac{{y}}{{{x}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}\frac{{y}}{{{x}}^{{3}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{3}\right]{,}{-}\frac{{1}}{{z}{}{x}{}{y}}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{2}\right]{,}\frac{{1}}{{z}{}{x}{}{y}}\right]\right]\right]\right)$ (2.8)

Calculate the Ricci tensor.

 M1 > Ric2 := RicciTensor(R2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{M1}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{y}}{{{x}}^{{3}}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{{1}}{{z}{}{x}{}{y}}\right]\right]\right]\right)$ (2.9)

Example 3.

In this example we calculate the Ricci tensor for a metric and note that in this case the Ricci tensor is symmetric.

 M1 > DGsetup([x, y, z], M);
 ${\mathrm{frame name: M}}$ (2.10)
 M > g := evalDG(y*dx &t dx + z*dy &t dy + dz &t dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{z}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (2.11)

Calculate the Ricci tensor for the metric directly.

 M > Ric3 := RicciTensor(g);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{4}{}{y}{}{z}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{4}}{}\frac{{{y}}^{{2}}{+}{z}}{{{y}}^{{2}}{}{z}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{4}{}{y}{}{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{4}{}{y}{}{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{1}}{{4}{}{{z}}^{{2}}}\right]\right]\right]\right)$ (2.12)