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Tensor[InvariantTensorsAtAPoint] - find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices

Calling Sequences

InvariantTensorsAtAPoint(A, S, options)

Parameters

A        - a list of square matrices, with dimension equal to the dimension of the space on which the tensors S are defined

S        - a list of tensors or differential forms, each of the same index type

options  - the keyword argument output

Description

 • This command calculates the tensors in the span of the tensors in the list S which are invariant with respect to the infinitesimal action generated by the matrices in the list A. This is a pointwise calculation.
 • Let  be the coordinates in terms of which the tensors in the list S are defined. If and then If   , then If  and are tensors, then . Thus, the action of $P$ on a tensor  defined at a point coincides with the Lie derivative of (as a tensor with constant coefficients) with respect to the linear vector fieldthat is, . See Example 6 for examples of this action of matrices on tensors.
 • If and then InvariantTensorsAtAPoint(A, S) returns a basis for the vector space of all tensors  (constant) such that
 • If no invariant tensors exist, an empty list is returned.
 • With output = "list", a list of invariant tensors is returned. This is the default. With output = "general", a single tensor with arbitrary coefficients is returned. If the number of matrices in the list A is 1 and output = "action", then the action of the matrix in A on the tensors in S is returned.
 • In many cases, the list of tensors to be used by InvariantTensorsAtAPoint can be created with the commands GenerateTensors, GenerateSymmetricTensors, GenerateForms.

Examples

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

Example 1.

Define a list of matrices for the first argument of InvariantTensorsAtAPoint .

 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right)\right]$

Define a 2-dimensional space on which the tensors $S$ for the second argument of InvariantTensorsAtAPoint will be defined.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

We take for the space of all rank 2 covariant tensors on $M$.

 M > $S≔\mathrm{evalDG}\left(\left[\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx},\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy},\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx},\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right]\right)$
 ${S}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}\right]$ (2.2)
 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 $\left[{-}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dy}}\right]$ (2.3)

Example 2.

Here we consider a simple example where the matrices depend upon the coordinates of the manifold on which the tensors are defined.

 M > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.4)
 > $A≔\left[\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[-1,0,0\right],\left[0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,1\right],\left[0,0,0\right],\left[-\frac{1}{{y}^{2}},0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,0\right],\left[0,0,1\right],\left[0,-\frac{1}{{y}^{2}},0\right]\right]\right)\right]$

We take for the space of all symmetric rank-2 covariant tensors on $M$.

 M > $S≔\mathrm{evalDG}\left(\left[\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx},\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy},\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz},\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy},\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz},\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right]\right)$
 ${S}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.5)

We find that the $A$-invariant tensors vary with the coordinate.

 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 $\left[\frac{{\mathrm{dx}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dy}}}{{{y}}^{{2}}}{+}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.6)

Example 3.

The classical simple Lie algebras can be defined as matrix algebras which leave a tensor or a collection of tensors invariant. In this example we check that the matrices defining the real sympletic algebra leave invariant a non-degenerate 2-form.

We first use the commands SimpleLieAlgebraData and StandardRepresentation to obtain the matrices defining .

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(4, R\right)",\mathrm{sp4R}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e4}}\right]$ (2.7)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sp4R}}$ (2.8)

Here are the 10 matrices for .

 sp4R > $A≔\mathrm{StandardRepresentation}\left(\mathrm{sp4R}\right)$

Let us find the 2-forms which are invariant with respect to these matrices. First define a 4-dimensional space.

 sp4R > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right):$

Generate a basis of 2-forms on

 V > $\mathrm{\Omega }≔\mathrm{Tools}:-\mathrm{GenerateForms}\left(\left[\mathrm{dx1},\mathrm{dx2},\mathrm{dx3},\mathrm{dx4}\right],2\right)$
 ${\mathrm{Ω}}{:=}\left[{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{,}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{,}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{,}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{,}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}{,}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}\right]$ (2.9)

The InvariantTensorsAtAPoint command shows that all 2-forms which are invariant with respect to the matrices are multiples of a single non-degenerate 2-form.

 V > $\mathrm{InvariantTensorsAtAPoint}\left(A,\mathrm{\Omega }\right)$
 $\left[{\mathrm{dx1}}{}{\mathrm{dx3}}{}{\bigwedge }{+}{\mathrm{dx2}}{}{\mathrm{dx4}}{}{\bigwedge }\right]$ (2.10)

Example 4.

The calculations of invariant tensors can be done in an anholonomic frame. (See FrameData.)

 V > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.11)
 M > $\mathrm{FD}≔\mathrm{FrameData}\left(\left[\mathrm{dx}+y\mathrm{dz},\mathrm{dy},\mathrm{dz}\right],N\right)$
 ${\mathrm{FD}}{:=}\left[{d}{}{\mathrm{Θ1}}{=}{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}{,}{d}{}{\mathrm{Θ2}}{=}{0}{,}{d}{}{\mathrm{Θ3}}{=}{0}\right]$ (2.12)
 M > $\mathrm{DGsetup}\left(\mathrm{FD},\left['\mathrm{X1}','\mathrm{X2}','\mathrm{X3}'\right],\left['\mathrm{ω1}','\mathrm{ω2}','\mathrm{ω3}'\right]\right)$
 ${\mathrm{frame name: N}}$ (2.13)
 N > $A≔\left[\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[0,0,1\right],\left[0,0,0\right]\right]\right)\right]$

Here is a basis for the A-invariant vectors.

 N > $B≔\left[\mathrm{X1},\mathrm{X2},\mathrm{X3}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.14)
 N > $\mathrm{InvariantTensorsAtAPoint}\left(A,B\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.15)

Here is a basis for the A-invariant 1-forms.

 N > $\mathrm{Ω1}≔\left[\mathrm{ω1},\mathrm{ω2},\mathrm{ω3}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.16)
 N > $\mathrm{InvariantTensorsAtAPoint}\left(A,\mathrm{Ω1}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.17)

Here is a basis for the A-invariant 2-forms.

 N > $\mathrm{Ω2}≔\mathrm{evalDG}\left(\left[\mathrm{ω1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{ω2},\mathrm{ω1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{ω3},\mathrm{ω2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{ω3}\right]\right)$
 ${\mathrm{Ω2}}{:=}\left[{\mathrm{ω1}}{}{\bigwedge }{}{\mathrm{ω2}}{,}{\mathrm{ω1}}{}{\bigwedge }{}{\mathrm{ω3}}{,}{\mathrm{ω2}}{}{\bigwedge }{}{\mathrm{ω3}}\right]$ (2.18)
 N > $\mathrm{InvariantTensorsAtAPoint}\left(A,\mathrm{Ω2}\right)$
 $\left[{\mathrm{ω2}}{}{\bigwedge }{}{\mathrm{ω3}}\right]$ (2.19)

Example 5.

One can use the command InvariantTensorsAtAPoint to calculate invariant tensors on the fiber of a bundle.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],\left[u,v\right],E\right)$
 ${\mathrm{frame name: E}}$ (2.20)
 E > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right)\right]$

Let be a basis for the space of symmetric rank 2 covariant tensors on the fiber of $E.$

 E > $S≔\mathrm{evalDG}\left(\left[\mathrm{D_u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_u},\mathrm{D_u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_v}+\mathrm{D_v}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_u},\mathrm{D_v}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_v}\right]\right)$
 ${S}{:=}\left[{\mathrm{D_u}}{}{\mathrm{D_u}}{,}{\mathrm{D_u}}{}{\mathrm{D_v}}{+}{\mathrm{D_u}}{}{\mathrm{D_v}}{,}{\mathrm{D_v}}{}{\mathrm{D_v}}\right]$ (2.21)
 E > $\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 $\left[{\mathrm{D_u}}{}{\mathrm{D_v}}{+}{\mathrm{D_u}}{}{\mathrm{D_v}}\right]$ (2.22)

Example 6.

With the keyword argument output = "action", we can obtain the action of a given matrix on a list of tensors.

 N > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.23)

 M > $A≔\left[\mathrm{Matrix}\left(\left[\left[a,b,c\right],\left[0,r,s\right],\left[0,0,t\right]\right]\right)\right]$
 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right],\mathrm{output}="action"\right)$
 $\left[{-}{a}{}{\mathrm{D_x}}{,}{-}{\mathrm{D_x}}{}{b}{-}{\mathrm{D_y}}{}{r}{,}{-}{\mathrm{D_x}}{}{c}{-}{\mathrm{D_y}}{}{s}{-}{\mathrm{D_z}}{}{t}\right]$ (2.24)
 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],\mathrm{output}="action"\right)$
 $\left[{a}{}{\mathrm{dx}}{+}{b}{}{\mathrm{dy}}{+}{c}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{r}{+}{\mathrm{dz}}{}{s}{,}{t}{}{\mathrm{dz}}\right]$ (2.25)
 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,\left[\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_y}\right],\mathrm{output}="action"\right)$
 $\left[{-}{b}{}{\mathrm{dx}}{}{\mathrm{D_x}}{+}\left({a}{-}{r}\right){}{\mathrm{dx}}{}{\mathrm{D_y}}{+}{b}{}{\mathrm{dy}}{}{\mathrm{D_y}}{+}{c}{}{\mathrm{dz}}{}{\mathrm{D_y}}\right]$ (2.26)

Example 7.

In this example we demonstrate how the command InvariantTensorsAtAPoint can be used in conjunction with InvariantGeometricObjectFields and IsotropySubalgebra to calculate tensors which are invariant with respect to a given infinitesimal group action.

The theory behind this example is as follows. If is a Lie algebra of vector fields on a manifold $M,$and  is a list of tensor fields on $M$, then the command InvariantGeometricObjectFields(Gamma, S) returns a basis for the tensor fields satisfying where  and the ${f}^{i}$ are functions on In situations where the vector fields in are algebraically complicated and/or the number of tensors in the listis large, it may take a very long time to calculate the invariant tensor fields. The command InvariantTensorsAtAPoint can be used to reduce the computation time by reducing the number of tensors in the list Let be the isotropy of at a point of Then for any matrix defining the linear isotropy representation of a vector in  one has that Consequently, one can replace the original list of tensor by the list of tensors which are invariant under the matrices defining the infinitesimal isotropy represention at a generic point. The infinitesimal isotropy representation of can be computed with the command IsotropySubalgebra in the GroupActions package.

 M > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.27)

We consider the following 6-dimensional infinitesimal group action, depending on a parameter $\mathrm{λ}.$

 M > $\mathrm{Gamma}≔\left[\frac{z\mathrm{cos}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)t\right)\mathrm{D_t}}{\mathrm{sqrt}\left(1+\mathrm{\lambda }{z}^{2}\right)}+\frac{\mathrm{sqrt}\left(1+\mathrm{\lambda }{z}^{2}\right)\mathrm{sin}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)t\right)\mathrm{D_z}}{\mathrm{sqrt}\left(\mathrm{\lambda }\right)},-\frac{z\mathrm{sin}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)t\right)\mathrm{D_t}}{\mathrm{sqrt}\left(1+\mathrm{\lambda }{z}^{2}\right)}+\frac{\mathrm{sqrt}\left(1+\mathrm{\lambda }{z}^{2}\right)\mathrm{cos}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)t\right)\mathrm{D_z}}{\mathrm{sqrt}\left(\mathrm{\lambda }\right)},-\frac{y\mathrm{cos}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)x\right)\mathrm{D_x}}{\mathrm{sqrt}\left(1-\mathrm{\lambda }{y}^{2}\right)}+\frac{\mathrm{sqrt}\left(1-\mathrm{\lambda }{y}^{2}\right)\mathrm{sin}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)x\right)\mathrm{D_y}}{\mathrm{sqrt}\left(\mathrm{\lambda }\right)},\frac{y\mathrm{sin}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)x\right)\mathrm{D_x}}{\mathrm{sqrt}\left(1-\mathrm{\lambda }{y}^{2}\right)}+\frac{\mathrm{sqrt}\left(1-\mathrm{\lambda }{y}^{2}\right)\mathrm{cos}\left(\mathrm{sqrt}\left(\mathrm{\lambda }\right)x\right)\mathrm{D_y}}{\mathrm{sqrt}\left(\mathrm{\lambda }\right)},-\frac{\mathrm{D_x}}{\mathrm{sqrt}\left(\mathrm{\lambda }\right)},-\frac{\mathrm{D_t}}{\mathrm{sqrt}\left(\mathrm{\lambda }\right)}\right]$
 ${\mathrm{Γ}}{:=}\left[\frac{{z}{}{\mathrm{cos}}{}\left(\sqrt{{\mathrm{λ}}}{}{t}\right){}{\mathrm{D_t}}}{\sqrt{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}}{+}\frac{\sqrt{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}{}{\mathrm{sin}}{}\left(\sqrt{{\mathrm{λ}}}{}{t}\right){}{\mathrm{D_z}}}{\sqrt{{\mathrm{λ}}}}{,}{-}\frac{{z}{}{\mathrm{sin}}{}\left(\sqrt{{\mathrm{λ}}}{}{t}\right){}{\mathrm{D_t}}}{\sqrt{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}}{+}\frac{\sqrt{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}{}{\mathrm{cos}}{}\left(\sqrt{{\mathrm{λ}}}{}{t}\right){}{\mathrm{D_z}}}{\sqrt{{\mathrm{λ}}}}{,}{-}\frac{{y}{}{\mathrm{cos}}{}\left(\sqrt{{\mathrm{λ}}}{}{x}\right){}{\mathrm{D_x}}}{\sqrt{{-}{{y}}^{{2}}{}{\mathrm{λ}}{+}{1}}}{+}\frac{\sqrt{{-}{{y}}^{{2}}{}{\mathrm{λ}}{+}{1}}{}{\mathrm{sin}}{}\left(\sqrt{{\mathrm{λ}}}{}{x}\right){}{\mathrm{D_y}}}{\sqrt{{\mathrm{λ}}}}{,}\frac{{y}{}{\mathrm{sin}}{}\left(\sqrt{{\mathrm{λ}}}{}{x}\right){}{\mathrm{D_x}}}{\sqrt{{-}{{y}}^{{2}}{}{\mathrm{λ}}{+}{1}}}{+}\frac{\sqrt{{-}{{y}}^{{2}}{}{\mathrm{λ}}{+}{1}}{}{\mathrm{cos}}{}\left(\sqrt{{\mathrm{λ}}}{}{x}\right){}{\mathrm{D_y}}}{\sqrt{{\mathrm{λ}}}}{,}{-}\frac{{\mathrm{D_x}}}{\sqrt{{\mathrm{λ}}}}{,}{-}\frac{{\mathrm{D_t}}}{\sqrt{{\mathrm{λ}}}}\right]$ (2.28)

Calculate the isotropy matrices at a generic point.

 M > $\mathrm{A0}≔\mathrm{IsotropySubalgebra}\left(\mathrm{Gamma},\left[t=\mathrm{t0},x=\mathrm{x0},y=\mathrm{y0},z=\mathrm{z0}\right],\mathrm{output}=\left["Representation"\right]\right)$

Since we are only interested in the span of these matrices, we can try to simplify the result using the command CanonicalBasis. We replace the fixed coordinates by their general values.

 M > $A≔\mathrm{subs}\left(\left[\mathrm{z0}=z,\mathrm{y0}=y\right],\mathrm{Tools}:-\mathrm{CanonicalBasis}\left(\mathrm{A0}\right)\right)$

Now we look for the $A-$invariant symmetric rank 2-tensors.

 M > $S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dt},\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],2\right)$
 ${S}{:=}\left[{\mathrm{dt}}{}{\mathrm{dt}}{,}\frac{{1}}{{2}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dt}}{,}\frac{{1}}{{2}}{}{\mathrm{dt}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dt}}{,}\frac{{1}}{{2}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dt}}{,}{\mathrm{dx}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.29)

Here is a basis for the isotropy invariant tensors.

 M > $\mathrm{InvTpt}≔\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 ${\mathrm{InvTpt}}{:=}\left[{-}\left({{z}}^{{4}}{}{{\mathrm{λ}}}^{{2}}{+}{2}{}{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}\right){}{\mathrm{dt}}{}{\mathrm{dt}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{,}\left({{y}}^{{4}}{}{{\mathrm{λ}}}^{{2}}{-}{2}{}{{y}}^{{2}}{}{\mathrm{λ}}{+}{1}\right){}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}\right]$ (2.30)

These tensors are not individually $\mathrm{Γ}$-invariant, but their span is $\mathrm{Γ}$-invariant, that is, the Lie derivative of these tensors with respect to each vector field in is a linear combination of the tensors $\mathrm{InvTpt}$. For example:

 M > $\mathrm{LieT}≔\mathrm{LieDerivative}\left(\mathrm{Gamma}\left[1\right],\mathrm{InvTpt}\right)$
 ${\mathrm{LieT}}{:=}\left[{-}\frac{{2}{}{z}{}{\mathrm{sin}}{}\left(\sqrt{{\mathrm{λ}}}{}{t}\right){}\sqrt{{\mathrm{λ}}}{}\left({{z}}^{{4}}{}{{\mathrm{λ}}}^{{2}}{+}{2}{}{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}\right){}{\mathrm{dt}}{}{\mathrm{dt}}}{\sqrt{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}}{+}\frac{{2}{}{z}{}{\mathrm{sin}}{}\left(\sqrt{{\mathrm{λ}}}{}{t}\right){}\sqrt{{\mathrm{λ}}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{\sqrt{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}}{,}{0}{}{\mathrm{dt}}{}{\mathrm{dt}}\right]$ (2.31)
 M > $\mathrm{GetComponents}\left(\mathrm{LieT},\mathrm{InvTpt}\right)$
 $\left[\left[\frac{{2}{}{z}{}{\mathrm{sin}}{}\left(\sqrt{{\mathrm{λ}}}{}{t}\right){}\sqrt{{\mathrm{λ}}}}{\sqrt{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}}{,}{0}\right]{,}\left[{0}{,}{0}\right]\right]$ (2.32)

Finally, we use the output of the InvariantTensorsAtAPoint command to calculate the $\mathrm{Γ}$-invariant tensors.

 M > $\mathrm{invT}≔\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma},\mathrm{InvTpt},\mathrm{output}="list"\right)$
 ${\mathrm{invT}}{:=}\left[\left({{y}}^{{2}}{}{\mathrm{λ}}{-}{1}\right){}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dy}}}{{{y}}^{{2}}{}{\mathrm{λ}}{-}{1}}{,}{-}\left({{z}}^{{2}}{}{\mathrm{λ}}{+}{1}\right){}{\mathrm{dt}}{}{\mathrm{dt}}{+}\frac{{\mathrm{dz}}{}{\mathrm{dz}}}{{{z}}^{{2}}{}{\mathrm{λ}}{+}{1}}\right]$ (2.33)

Note that the $\mathrm{Γ}$-invariant tensors are simply multiples of the isotropy invariant tensors. We check our final result.

 M > $\mathrm{LieDerivative}\left(\mathrm{Gamma},\mathrm{invT}\right)$
 $\left[\left[{0}{}{\mathrm{dt}}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}{}{\mathrm{dt}}\right]{,}\left[{0}{}{\mathrm{dt}}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}{}{\mathrm{dt}}\right]{,}\left[{0}{}{\mathrm{dt}}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}{}{\mathrm{dt}}\right]{,}\left[{0}{}{\mathrm{dt}}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}{}{\mathrm{dt}}\right]{,}\left[{0}{}{\mathrm{dt}}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}{}{\mathrm{dt}}\right]{,}\left[{0}{}{\mathrm{dt}}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}{}{\mathrm{dt}}\right]\right]$ (2.34)

Example 8.

In this example we demonstrate how the command InvariantTensorsAtAPoint can be used in conjunction with CovariantlyConstantTensors and IsotropySubalgebra to calculate tensors which are covariantly constant.

The theory underlying this example is as follows. If  is an affine connection  and a list of tensor fields on $M,$then the command CovariantlyConstantTensors($\nabla$, S) returns a basis for the tensor fields satisfying where  and the ${f}^{i}$ are functions on If  ${H}_{x}$ is a list of matrices defining the infinitesimal holonomy of the connection $\nabla$ at a point  in, then the covariantly constant tensor $T$ satisfies for every matrix Consequently, one can replace the original list of tensors by the list of tensors which are invariant under the matrices defining the infinitesimal holonomy at a generic point. The infinitesimal holonomyof the connection can be computed with the command InfinitesimalHolonomy in the Tensor package.

We use the split signature Fubini-Study metric in four-dimensions. (The complex change of variables gives the usual Riemannian metric):

 P > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.35)

Introduce a symmetric tensor and two 1-forms which will be used to define the metric.

 P > $n≔\mathrm{z1}\mathrm{w1}+\mathrm{z2}\mathrm{w2}$
 ${n}{:=}{\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}$ (2.36)
 P > $\mathrm{o1}≔\mathrm{evalDG}\left(\mathrm{dz1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}+\mathrm{dz2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)$
 ${\mathrm{o1}}{:=}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{+}\frac{{1}}{{2}}{}{\mathrm{dw1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dw2}}{}{\mathrm{dz2}}$ (2.37)
 P > $\mathrm{o2}≔\mathrm{evalDG}\left(\mathrm{z1}\mathrm{dw1}+\mathrm{z2}\mathrm{dw2}\right)$
 ${\mathrm{o2}}{:=}{\mathrm{dw1}}{}{\mathrm{z1}}{+}{\mathrm{dw2}}{}{\mathrm{z2}}$ (2.38)
 P > $\mathrm{o3}≔\mathrm{evalDG}\left(\mathrm{w1}\mathrm{dz1}+\mathrm{w2}\mathrm{dz2}\right)$
 ${\mathrm{o3}}{:=}{\mathrm{dz1}}{}{\mathrm{w1}}{+}{\mathrm{dz2}}{}{\mathrm{w2}}$ (2.39)

Here is the metric we shall use.

 P > $g≔\mathrm{evalDG}\left(\frac{1}{{\left(1+n\right)}^{2}}\left(\left(1+n\right)\mathrm{o1}-\mathrm{o2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{o3}-\mathrm{o3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{o2}\right)\right)$
 ${g}{:=}{-}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{-}{1}\right){}{\mathrm{dz1}}{}{\mathrm{dw1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z2}}{}{\mathrm{w1}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z1}}{}{\mathrm{w2}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right){}{\mathrm{dz2}}{}{\mathrm{dw2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{-}{1}\right){}{\mathrm{dw1}}{}{\mathrm{dz1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z1}}{}{\mathrm{w2}}{}{\mathrm{dw1}}{}{\mathrm{dz2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z2}}{}{\mathrm{w1}}{}{\mathrm{dw2}}{}{\mathrm{dz1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right){}{\mathrm{dw2}}{}{\mathrm{dz2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}$ (2.40)

We calculate the infinitesimal holonomy for the metric $g$ at a generic point. We simplify the result with the CanonicalBasis command.

 P > $\mathrm{H0}≔\mathrm{InfinitesimalHolonomy}\left(g,\left[\right]\right):$
 P > $A≔\mathrm{Tools}:-\mathrm{CanonicalBasis}\left(%\right)$

First we find the symmetric rank 2 tensors which areinvariant with respect to the infinitesimal holonomy.

 P > $S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dz1},\mathrm{dz2},\mathrm{dw1},\mathrm{dw2}\right],2\right)$
 ${S}{:=}\left[{\mathrm{dz1}}{}{\mathrm{dz1}}{,}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{,}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{+}\frac{{1}}{{2}}{}{\mathrm{dw1}}{}{\mathrm{dz1}}{,}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{+}\frac{{1}}{{2}}{}{\mathrm{dw2}}{}{\mathrm{dz1}}{,}{\mathrm{dz2}}{}{\mathrm{dz2}}{,}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{+}\frac{{1}}{{2}}{}{\mathrm{dw1}}{}{\mathrm{dz2}}{,}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{+}\frac{{1}}{{2}}{}{\mathrm{dw2}}{}{\mathrm{dz2}}{,}{\mathrm{dw1}}{}{\mathrm{dw1}}{,}\frac{{1}}{{2}}{}{\mathrm{dw1}}{}{\mathrm{dw2}}{+}\frac{{1}}{{2}}{}{\mathrm{dw2}}{}{\mathrm{dw1}}{,}{\mathrm{dw2}}{}{\mathrm{dw2}}\right]$ (2.41)

We find the only symmetric rank-2 tensor which is covariantly constant is a constant multiple of the metric.

 P > $\mathrm{InvariantS}≔\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 ${\mathrm{InvariantS}}{:=}\left[{-}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{-}{1}\right){}{\mathrm{dz1}}{}{\mathrm{dw1}}}{{\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}}{-}\frac{{\mathrm{w1}}{}{\mathrm{z2}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}}{{\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}}{-}\frac{{\mathrm{w2}}{}{\mathrm{z1}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}}{{\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{-}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{-}{1}\right){}{\mathrm{dw1}}{}{\mathrm{dz1}}}{{\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}}{-}\frac{{\mathrm{w2}}{}{\mathrm{z1}}{}{\mathrm{dw1}}{}{\mathrm{dz2}}}{{\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}}{-}\frac{{\mathrm{w1}}{}{\mathrm{z2}}{}{\mathrm{dw2}}{}{\mathrm{dz1}}}{{\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}}{+}\frac{{1}}{{2}}{}{\mathrm{dw2}}{}{\mathrm{dz2}}\right]$ (2.42)
 M > $\mathrm{CovariantlyConstantTensors}\left(g,\mathrm{InvariantS}\right)$
 $\left[{-}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{-}{1}\right){}{\mathrm{dz1}}{}{\mathrm{dw1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z2}}{}{\mathrm{w1}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z1}}{}{\mathrm{w2}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right){}{\mathrm{dz2}}{}{\mathrm{dw2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{-}{1}\right){}{\mathrm{dw1}}{}{\mathrm{dz1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z1}}{}{\mathrm{w2}}{}{\mathrm{dw1}}{}{\mathrm{dz2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{-}\frac{{\mathrm{z2}}{}{\mathrm{w1}}{}{\mathrm{dw2}}{}{\mathrm{dz1}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{\left({\mathrm{w1}}{}{\mathrm{z1}}{-}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right){}{\mathrm{dw2}}{}{\mathrm{dz2}}}{{\left({\mathrm{w1}}{}{\mathrm{z1}}{+}{\mathrm{w2}}{}{\mathrm{z2}}{+}{1}\right)}^{{2}}}\right]$ (2.43)

Next we look for the (1,1) tensors which are invariant with respect to the infinitesimal holonomy.

 P > $J≔\mathrm{GenerateTensors}\left(\left[\left[\mathrm{D_z1},\mathrm{D_z2},\mathrm{D_w1},\mathrm{D_w2}\right],\left[\mathrm{dz1},\mathrm{dz2},\mathrm{dw1},\mathrm{dw2}\right]\right]\right)$
 ${J}{:=}\left[{\mathrm{D_z1}}{}{\mathrm{dz1}}{,}{\mathrm{D_z1}}{}{\mathrm{dz2}}{,}{\mathrm{D_z1}}{}{\mathrm{dw1}}{,}{\mathrm{D_z1}}{}{\mathrm{dw2}}{,}{\mathrm{D_z2}}{}{\mathrm{dz1}}{,}{\mathrm{D_z2}}{}{\mathrm{dz2}}{,}{\mathrm{D_z2}}{}{\mathrm{dw1}}{,}{\mathrm{D_z2}}{}{\mathrm{dw2}}{,}{\mathrm{D_w1}}{}{\mathrm{dz1}}{,}{\mathrm{D_w1}}{}{\mathrm{dz2}}{,}{\mathrm{D_w1}}{}{\mathrm{dw1}}{,}{\mathrm{D_w1}}{}{\mathrm{dw2}}{,}{\mathrm{D_w2}}{}{\mathrm{dz1}}{,}{\mathrm{D_w2}}{}{\mathrm{dz2}}{,}{\mathrm{D_w2}}{}{\mathrm{dw1}}{,}{\mathrm{D_w2}}{}{\mathrm{dw2}}\right]$ (2.44)
 P > $\mathrm{InvariantJ}≔\mathrm{InvariantTensorsAtAPoint}\left(A,J\right)$
 ${\mathrm{InvariantJ}}{:=}\left[{\mathrm{dw1}}{}{\mathrm{D_w1}}{+}{\mathrm{dw2}}{}{\mathrm{D_w2}}{+}{\mathrm{dz1}}{}{\mathrm{D_z1}}{+}{\mathrm{dz2}}{}{\mathrm{D_z2}}\right]$ (2.45)

There are two type (1,1) tensors which areinvariant with respect to the infinitesimal holonomy and both of these are actually covariantly constant.

 P > $\mathrm{CovariantlyConstantTensors}\left(g,\mathrm{InvariantJ}\right)$
 $\left[{\mathrm{dw1}}{}{\mathrm{D_w1}}{+}{\mathrm{dw2}}{}{\mathrm{D_w2}}{+}{\mathrm{dz1}}{}{\mathrm{D_z1}}{+}{\mathrm{dz2}}{}{\mathrm{D_z2}}\right]$ (2.46)