Let $p\in M$ and let $S$ be a $k$-dimensional subspace of the tangent space $T_{p}M\.$ Let $\left(Y_1\,Y_2\, ..\.\, Y_k\right)$be a basis for $S$ and let $h$ be the $k\times k$ matrix with components

Let $S$ be the span of $m$ vectors $\left(X_1\,X_2\,..\.\,X_m\right)$; these vectors need not be linearly independent. The command SubspaceType(g, S) will attempt to determine the type of the subspace $S$ at a generic point. It will return a sequence k, T, where k is the dimension of $S$ and T is one of {"Riemannian", "PseudoRiemannian", "Null", Fail}. The value Fail typically indicates the subspace type is varying with respect to coordinates or parameters.

With the optional argument pt, the type of the subspace $S$ at pt is determined. Specific properties of the coordinates of the point pt can be specified using assuming.

With the keyword argument output = "Matrix", the values of the matrix $h$ and the basis used to compute $h$, evaluated at the point $p$, are returned.

With the keyword argument output = "Riemannian", output = "PseudoRiemannian" or output = "Null", the command returns the sequence k, TF, where k is the dimension of the subspace $S$ and TF is true or false.

The user can trace the program branching by setting infolevel[SubspaceType] := 2.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SubspaceType(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-SubspaceType.

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

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

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{S1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{S1}:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{SubspaceType}\left(g\,\mathrm{S1}\right)$

$2\,''Riemannian''$

$\mathrm{S2}≔\left[\mathrm{D\_t}\,\mathrm{D\_x}\right]$

$\mathrm{S2}:=\left[\mathrm{D\_t}\,\mathrm{D\_x}\right]$

$\mathrm{SubspaceType}\left(g\,\mathrm{S2}\right)$

$2\,''PseudoRiemannian''$

$\mathrm{S3}≔\left[\mathrm{D\_t}-\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{S3}:=\left[\mathrm{D\_t}-\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{SubspaceType}\left(g\,\mathrm{S3}\right)$

$2\,''Null''$

$S≔\left[t\mathrm{D\_t}-x\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$S:=\left[\mathrm{D\_t}t-\mathrm{D\_x}x\,\mathrm{D\_y}\right]$

$\mathrm{SubspaceType}\left(g\,S\right)$

$2\,\mathrm{Fail}$

$\mathrm{SubspaceType}\left(g\,S\,\left[t=a\,x=b\,y=c\,z=d\right]\,\mathrm{output}=''Matrix''\right)$

$\mathrm{SubspaceType}\left(g\,S\,\left[t=a\,x=a\,y=c\,z=d\right]\right)$

$2\,''Null''$

$\mathrm{SubspaceType}\left(g\,S\,\left[t=a\,x=-a\,y=c\,z=d\right]\right)$

$2\,''Null''$

$\mathrm{SubspaceType}\left(g\&comma;S\&comma;\left[t=a\&comma;x=b\&comma;y=c\&comma;z=d\right]\right)\mathbf{assuming}{b}^2<a^2$

$2\&comma;''PseudoRiemannian''$

$\mathrm{SubspaceType}\left(g\&comma;S\&comma;\left[t=a\&comma;x=b\&comma;y=c\&comma;z=d\right]\right)\mathbf{assuming}{a}^2<b^2$

$2\&comma;''Riemannian''$

$\mathrm{SubspaceType}\left(g\&comma;\mathrm{S2}\&comma;\mathrm{output}=''Null''\right)$

$2\&comma;\mathrm{false}$

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

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

$\mathrm{SubspaceType}\left(g\&comma;\mathrm{S2}\right)$

The orbit type is "Null" if the induced metric on the orbit is degenerate
The orbit type is "Riemannian" if the induced metric on the orbit is positive or negative definite
The orbit type is "PseudoRiemannian" if the induced metric on the orbit is non-degenerate and indefinite
   The matrix h defining the induced metric is:

   Matrix(2, 2, [[1,0],[0,-1]])
The determinant of the matrix h is: -1

$2\&comma;''PseudoRiemannian''$

$\mathrm{infolevel}\left[\mathrm{SubspaceType}\right]≔0$

${\mathrm{infolevel}}_{\mathrm{DifferentialGeometry:-Tensor:-SubspaceType}}:=0$

$\mathrm{DGsetup}\left(\left[u\&comma;v\&comma;t\&comma;x\&comma;y\&comma;z\right]\&comma;M\right)$

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

$g≔\mathrm{evalDG}\left(\mathrm{du}\&s\mathrm{dv}+\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\frac{1}{2}\mathrm{du}\mathrm{dv}+\frac{1}{2}\mathrm{dv}\mathrm{du}+\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{S1}≔\left[\mathrm{D\_u}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\right]$

$\mathrm{S1}:=\left[\mathrm{D\_u}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\right]$

$\mathrm{SubspaceType}\left(g\&comma;\mathrm{S1}\right)$

The orbit type is "Null" if the induced metric on the orbit is degenerate

The orbit type is "Riemannian" if the induced metric on the orbit is positive or negative definite
The orbit type is "PseudoRiemannian" if the induced metric on the orbit is non-degenerate and indefinite
   The matrix h defining the induced metric is:
   Matrix(3, 3, [[0,0,0],[0,-1,0],[0,0,-1]])
The determinant of the matrix h is: 0

$3\&comma;''Null''$

$\mathrm{S2}≔\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$\mathrm{S2}:=\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$\mathrm{SubspaceType}\left(g\&comma;\mathrm{S2}\right)$

The orbit type is "Null" if the induced metric on the orbit is degenerate

The orbit type is "Riemannian" if the induced metric on the orbit is positive or negative definite
The orbit type is "PseudoRiemannian" if the induced metric on the orbit is non-degenerate and indefinite
   The matrix h defining the induced metric is:
   Matrix(4, 4, [[1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1]])
The determinant of the matrix h is: -1

$4\&comma;''PseudoRiemannian''$

$\mathrm{S3}≔\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;y\mathrm{D\_x}-x\mathrm{D\_y}\right]$

$\mathrm{S3}:=\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;y\mathrm{D\_x}-x\mathrm{D\_y}\right]$

$\mathrm{SubspaceType}\left(g\&comma;\mathrm{S3}\right)$

$2\&comma;''Riemannian''$

$\mathrm{SubspaceType}\left(g\&comma;\mathrm{S3}\&comma;\mathrm{output}=''Matrix''\right)$