Let $\mathrm{\Γ}$ be a connection on a vector bundle $\mathrm{\π} \: E \to M$ with associated directional covariant derivative operator $\nabla$and curvature tensor $R$. For any point $p\in M$and vector fields $X\, Y\, Z_1\, Z_2\, ..\. \,Z_{\mathrm{\ℓ}}$ , the curvature tensor and its directional covariant derivatives can be viewed as linear maps

The command InfinitesimalHolonomy returns a list of holonomy matrices for the infinitesimal holonomy of the given metric or connection. These matrices are guaranteed to be linearly independent at the point with the coordinates specified by the second argument initpt. At a generically chosen point these matrices will generate the full infinitesimal holonomy algebra in a neighborhood of that point. The structure equations for the Lie algebra defined by these matrices can be determined by the command LieAlgebraData.

The keyword arguments are used as follows. The keyword argument auxilaryequations specifies a set of equations which are to be used to simplify the connection, the curvature tensor and its covariant derivatives. If a certain number or holonomy matrices are known (perhaps by a previous computation) then these can be included as part of the infinitesimal holonomy algebra with the keyword argument knownholonomy. The computations can be terminated at a given derivative order of the curvature tensor with keyword argument curvatureorder. The computations can be terminated at a given holonomy dimension with keyword argument maximumholonomy. With output = "tensor", the holonomy matrices are represented by type (1,1) tensors.

The number of holonomy matrices calculated at each curvature order can be monitored by setting infolevel[ InfinitesimalHolonomy] := 2. With the infolevel set to 3, the holonomy matrices calculated at each curvature order are displayed (as lists of lists).

The curvature tensor $R\left(x\right)$ of $\mathrm{\Γ}$ is computed and a list $\mathrm{\𝒜}\left(x\right)$of type (1,1) tensors$$spanning the linear maps $R\left(x\right)\left(X\, Y\right)$as $X\, Y$vary is determined. (The tensors in the list$\mathit{ }\mathrm{\𝒜}\left(x\right)$are functions of the coordinates $x$used to create the manifold $M$.)

Let ${\mathrm{\𝒜}}_0 \=$$\mathrm{\𝒜}\left(x_0\right)\mathit{ }$denote the tensors in the list $\mathrm{\𝒜}\left(x\right)$ evaluated at the given point $x_0\.$A maximally independent list of tensors $\mathrm{\ℬ}\mathit{ }\subset$${\mathrm{\𝒜}}_0$ is chosen using the DGbasis command. Let $n_0$ be the number of elements of the list $ℬ$. Each element of $ℬ$ is the restriction of an element of $𝒜\left(x\right)$. Let$\mathit{ }{\mathrm{\ℋ}}_0\left(x_{}\right)\subset \mathrm{\𝒜}\left(x\right)$ be the list of those $n_0$tensors whose restrictions define $\mathrm{\ℬ}$. We are guaranteed that [i] the $n_0$tensors in the list$\mathit{ }{\mathrm{\ℋ}}_0\left(x_{}\right)$are pointwise independent in a neighborhood of $x_0$; [ii] that each of these tensors defines an element of infinitesimal holonomy at each point of this neighborhood of $x_0\;$and [iii] that the pointwise dimension of $\mathrm{\𝒜}$$\left(x\right)\ge n_0$ ($\mathbf{†}\mathbf{\)}$.

The directional derivatives of the tensors ${\mathrm{\ℋ}}_0\left(x_{}\right)$, in all coordinate directions, are computed to generate a new list of (1,1) tensors ${\mathrm{\𝒜}}_1\left(x_{}\right)$. The process from the previous step is repeated, as follows. Let ${\mathrm{\𝒜}}_{1\,0}\=$${\mathrm{\𝒜}}_1\left(x_0\right)\mathit{ }$and choose a maximal list of tensors ${\mathrm{\ℬ}}_1\subset$${\mathrm{\𝒜}}_{1\,0}$ such that the tensors ${\mathrm{\ℬ}}_1 \cup$ ${\mathrm{\ℋ}}_0\left(x_0\right)$are linearly independent. Again this gives $n_1$tensors ${\mathscr{H}}_1\left(x_{}\right)\subset {\mathrm{\𝒜}}_1\left(x\right)$ so that ${\mathrm{\ℬ}}_1$$\mathit{ }\mathit{\=}{\mathrm{\ℋ}}_1\left(x_0\right)$. We are guaranteed that the $n_0\+ n_{1 }$tensors$\mathit{ }{\mathrm{\ℋ}}_0\left(x_{}\right)$$\cup$ ${\mathrm{\ℋ}}_1\left(x_{}\right)$are [i] pointwise independent in a neighborhood of $x_0$; [ii] that each of these tensors defines an element of infinitesimal holonomy at each point of this neighborhood of $x_0\;$and [iii] that the pointwise dimension of ${\mathrm{\𝒜}}_{}\left(x\right)$$\cup$ ${\mathrm{\𝒜}}_1\left(x_{}\right) \ge n_0\+ n_1$. ($\mathbf{‡}$)

The directional derivatives of the tensors ${\mathrm{\ℋ}}_1\left(x_{}\right)$are computed and the previous step repeated. The algorithm terminates if [i] no new holonomy elements are added at a given step, or [ii] if a fixed number of curvature derivatives (as specified by curvatureorder) are calculated.

The output of the command InfinitesimalHolonomy is a list of pointwise independent matrices or (1,1) tensors depending upon the coordinates $x$, each of which defines an element of the infinitesimal holonomy algebra at $x$. When the inequalities$\left(\mathbf{†}\right)$and $\left(\mathbf{‡}\right)$ are equalities the output of InfinitesimalHolonomy is a basis for the full infinitesimal holonomy algebra at $x$.

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

$\mathrm{DGsetup}\left(\left[x\,y\,z\,u\,v\right]\,\mathrm{M1}\right)$

$\mathrm{frame\; name:\; M1}$

$\mathrm{g1}≔\mathrm{evalDG}\left(x^3\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)+u^3\left(\mathrm{du}\&t\mathrm{du}+\mathrm{dv}\&t\mathrm{dv}\right)\right)$

$\mathrm{g1}:=x^3\mathrm{dx}\mathrm{dx}+x^3\mathrm{dy}\mathrm{dy}+x^3\mathrm{dz}\mathrm{dz}+u^3\mathrm{du}\mathrm{du}+u^3\mathrm{dv}\mathrm{dv}$

$\mathrm{InfHol1}≔\mathrm{InfinitesimalHolonomy}\left(\mathrm{g1}\,\left[x=1\,y=0\,z=1\,u=1\,v=1\right]\right)$

$\mathrm{InfHol1a}≔\mathrm{Tools}:-\mathrm{CanonicalBasis}\left(\mathrm{InfHol1}\right)$

$\mathrm{InfHol1}≔\mathrm{InfinitesimalHolonomy}\left(\mathrm{g1}\,\left[x=1\,y=0\,z=1\,u=1\,v=1\right]\,\mathrm{output}=''tensor''\right)$

$\mathrm{InfHol1}:=\left[\frac{3}{2x^2}\mathrm{D\_x}\mathrm{dy}-\frac{3}{2x^2}\mathrm{D\_y}\mathrm{dx}\,\frac{3}{2x^2}\mathrm{D\_x}\mathrm{dz}-\frac{3}{2x^2}\mathrm{D\_z}\mathrm{dx}\,-\frac{9}{4x^2}\mathrm{D\_y}\mathrm{dz}+\frac{9}{4x^2}\mathrm{D\_z}\mathrm{dy}\,\frac{3}{2u^2}\mathrm{D\_u}\mathrm{dv}-\frac{3}{2u^2}\mathrm{D\_v}\mathrm{du}\right]$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,\mathrm{M2}\right)$

$\mathrm{frame\; name:\; M2}$

$\mathrm{g2}≔\mathrm{evalDG}\left(t^a\mathrm{dx}\&t\mathrm{dx}+t^b\mathrm{dy}\&t\mathrm{dy}+t^c\mathrm{dz}\&t\mathrm{dz}-\mathrm{dt}\&t\mathrm{dt}\right)$

$\mathrm{g2}:=-\mathrm{dt}\mathrm{dt}+t^a\mathrm{dx}\mathrm{dx}+t^b\mathrm{dy}\mathrm{dy}+t^c\mathrm{dz}\mathrm{dz}$

$\mathrm{infolevel}\left[\mathrm{InfinitesimalHolonomy}\right]≔2$

${\mathrm{infolevel}}_{\mathrm{DifferentialGeometry:-Tensor:-InfinitesimalHolonomy}}:=2$

$\mathrm{InfHol2}≔\mathrm{InfinitesimalHolonomy}\left(\mathrm{g2}\,\left[t=1\right]\right)$

Calculating the derivatives of curvature at order 0

     The dimension of the holonomy algebra at order 0 is 6

$\mathrm{simplify}\left(\mathrm{Tools}:-\mathrm{CanonicalBasis}\left(\mathrm{InfHol2}\right)\right)$

$\mathrm{g2a}≔\mathrm{evalDG}\left(\mathrm{subs}\left(\left[a=2\,b=2\,c=2\right]\,t^2\mathrm{dx}\&t\mathrm{dx}+t^b\mathrm{dy}\&t\mathrm{dy}+t^c\mathrm{dz}\&t\mathrm{dz}-\mathrm{dt}\&t\mathrm{dt}\right)\right)$

$\mathrm{g2a}:=-\mathrm{dt}\mathrm{dt}+t^2\mathrm{dx}\mathrm{dx}+t^2\mathrm{dy}\mathrm{dy}+t^2\mathrm{dz}\mathrm{dz}$

$\mathrm{infolevel}\left[\mathrm{InfinitesimalHolonomy}\right]≔3$

${\mathrm{infolevel}}_{\mathrm{DifferentialGeometry:-Tensor:-InfinitesimalHolonomy}}:=3$

$\mathrm{InfHol2a}≔\mathrm{InfinitesimalHolonomy}\left(\mathrm{g2a}\,\left[t=1\right]\right)$

Calculating the derivatives of curvature at order 0

     The dimension of the holonomy algebra at order 0 is 3
     Matrix 1: [[0, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [0, 0, 0, 0]]
     Matrix 2: [[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, -1, 0, 0]]
     Matrix 3: [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]]
Calculating the derivatives of curvature at order 1

     The dimension of the holonomy algebra at order 1 is 6
     Matrix 1: [[0, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [0, 0, 0, 0]]
     Matrix 1: [[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, -1, 0, 0]]
     Matrix 1: [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]]
     Matrix 1: [[0, 0, t, 0], [0, 0, 0, 0], [1/t, 0, 0, 0], [0, 0, 0, 0]]
     Matrix 1: [[0, -t, 0, 0], [-1/t, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
     Matrix 1: [[0, 0, 0, t], [0, 0, 0, 0], [0, 0, 0, 0], [1/t, 0, 0, 0]]

$\mathrm{DGsetup}\left(\left[x\,y\,u\,v\right]\,\mathrm{M3}\right)\:$

$\mathrm{g3}≔\mathrm{evalDG}\left(C\left(u\right)\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)+\mathrm{du}\&s\mathrm{dv}\right)$

$\mathrm{g3}:=C\left(u\right)\mathrm{dx}\mathrm{dx}+C\left(u\right)\mathrm{dy}\mathrm{dy}+\frac{1}{2}\mathrm{du}\mathrm{dv}+\frac{1}{2}\mathrm{dv}\mathrm{du}$

$\mathrm{InfHol3}≔\mathrm{InfinitesimalHolonomy}\left(\mathrm{g3}\,\left[u=1\right]\,\mathrm{auxiliaryequations}=\left[\mathrm{diff}\left(C\left(u\right)\,\mathrm{`\$`}\left(u\,2\right)\right)=\frac{b}{u^2}C\left(u\right)\right]\right)$

Calculating the derivatives of curvature at order 0

     The dimension of the holonomy algebra at order 0 is 2
     Matrix 1: [[0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0, 0]]
     Matrix 2: [[0, 0, 0, 0], [0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0]]
Calculating the derivatives of curvature at order 1

     The dimension of the holonomy algebra at order 1 is 2
     Matrix 1: [[0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0, 0]]
     Matrix 1: [[0, 0, 0, 0], [0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0]]

$\mathrm{Tools}:-\mathrm{CanonicalBasis}\left(\mathrm{InfHol3}\right)$

$\mathrm{InfHol3}≔\mathrm{InfinitesimalHolonomy}\left(\mathrm{g3}\,\left[u=1\right]\,\mathrm{auxiliaryequations}=\left[\mathrm{diff}\left(C\left(u\right)\,\mathrm{`\$`}\left(u\,2\right)\right)=\frac{b}{u^2}C\left(u\right)\,\mathrm{eval}\left(\mathrm{diff}\left(C\left(u\right)\,u\right)\,u=1\right)=\mathrm{\alpha }\,C\left(1\right)=\mathrm{\beta }\right]\,\mathrm{holonomyatpoint}=''yes''\right)$

Calculating the derivatives of curvature at order 0

     The dimension of the holonomy algebra at order 0 is 2
     Matrix 1: [[0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0, 0]]
     Matrix 2: [[0, 0, 0, 0], [0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0]]
Calculating the derivatives of curvature at order 1

     The dimension of the holonomy algebra at order 1 is 2
     Matrix 1: [[0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0, 0]]
     Matrix 1: [[0, 0, 0, 0], [0, 0, -1/4*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u)^2, 0], [0, 0, 0, 0], [0, 1/2*(-diff(C(u),u)^2*u^2+2*b*C(u)^2)/u^2/C(u), 0, 0]]

$\mathrm{g1}$

$x^3\mathrm{dx}\mathrm{dx}+x^3\mathrm{dy}\mathrm{dy}+x^3\mathrm{dz}\mathrm{dz}+u^3\mathrm{du}\mathrm{du}+u^3\mathrm{dv}\mathrm{dv}$

$\mathrm{InfHol1a}$

$\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\mathrm{InfHol1a}\,\mathrm{hol1}\right)$

$\mathrm{LD}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=-\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; hol1}$

$\mathrm{Query}\left(''Indecomposable''\right)$

$\mathrm{false}$

$C≔\mathrm{Christoffel}\left(\mathrm{g1}\right)$

$C:=\frac{3}{2x}\mathrm{D\_x}\mathrm{dx}\mathrm{dx}-\frac{3}{2x}\mathrm{D\_x}\mathrm{dy}\mathrm{dy}-\frac{3}{2x}\mathrm{D\_x}\mathrm{dz}\mathrm{dz}+\frac{3}{2x}\mathrm{D\_y}\mathrm{dx}\mathrm{dy}+\frac{3}{2x}\mathrm{D\_y}\mathrm{dy}\mathrm{dx}+\frac{3}{2x}\mathrm{D\_z}\mathrm{dx}\mathrm{dz}+\frac{3}{2x}\mathrm{D\_z}\mathrm{dz}\mathrm{dx}+\frac{3}{2u}\mathrm{D\_u}\mathrm{du}\mathrm{du}-\frac{3}{2u}\mathrm{D\_u}\mathrm{dv}\mathrm{dv}+\frac{3}{2u}\mathrm{D\_v}\mathrm{du}\mathrm{dv}+\frac{3}{2u}\mathrm{D\_v}\mathrm{dv}\mathrm{du}$

$Q≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\,\mathrm{du}\,\mathrm{dv}\right]\,2\right)$

$Q:=\left[\mathrm{dx}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}+\frac{1}{2}\mathrm{dy}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{dz}+\frac{1}{2}\mathrm{dz}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{du}+\frac{1}{2}\mathrm{du}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{dv}+\frac{1}{2}\mathrm{dv}\mathrm{dx}\,\mathrm{dy}\mathrm{dy}\,\frac{1}{2}\mathrm{dy}\mathrm{dz}+\frac{1}{2}\mathrm{dz}\mathrm{dy}\,\frac{1}{2}\mathrm{dy}\mathrm{du}+\frac{1}{2}\mathrm{du}\mathrm{dy}\,\frac{1}{2}\mathrm{dy}\mathrm{dv}+\frac{1}{2}\mathrm{dv}\mathrm{dy}\,\mathrm{dz}\mathrm{dz}\,\frac{1}{2}\mathrm{dz}\mathrm{du}+\frac{1}{2}\mathrm{du}\mathrm{dz}\,\frac{1}{2}\mathrm{dz}\mathrm{dv}+\frac{1}{2}\mathrm{dv}\mathrm{dz}\,\mathrm{du}\mathrm{du}\,\frac{1}{2}\mathrm{du}\mathrm{dv}+\frac{1}{2}\mathrm{dv}\mathrm{du}\,\mathrm{dv}\mathrm{dv}\right]$

$\mathrm{CovariantlyConstantTensors}\left(C\,Q\right)$

$\left[u^3\mathrm{du}\mathrm{du}+u^3\mathrm{dv}\mathrm{dv}\,x^3\mathrm{dx}\mathrm{dx}+x^3\mathrm{dy}\mathrm{dy}+x^3\mathrm{dz}\mathrm{dz}\right]$

$\mathrm{InvQ}≔\mathrm{InvariantTensorsAtAPoint}\left(\mathrm{InfHol1a}\,Q\right)$

$\mathrm{InvQ}:=\left[\mathrm{du}\mathrm{du}\+\mathrm{dv}\mathrm{dv}\,\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}\right]$

$\mathrm{CovariantlyConstantTensors}\left(C\,\mathrm{InvQ}\right)$

$\left[x^3\mathrm{dx}\mathrm{dx}\+x^3\mathrm{dy}\mathrm{dy}\+x^3\mathrm{dz}\mathrm{dz}\,u^3\mathrm{du}\mathrm{du}\+u^3\mathrm{dv}\mathrm{dv}\right]$

$\mathrm{nops}\left(Q\right)$

$15$