HomothetyVectors - Maple Help

Tensor[HomothetyVectors] - calculate the homothety vectors for a given metric

Calling Sequences

HomothetyVectors( g, options)

Parameters

g         - a metric tensor on a manifold

options   - any of the following keywords arguments: ansatz, unknowns, auxiliaryequations, coefficientvariables, parameters, output

Description

 • A vector field is called a homothety vector for the metric $g$ if, where denotes Lie differentiation with respect to and $\mathrm{λ}$ is a constant (proportional to the divergence of $X$). If  and $▿$denotes covariant differentiation with respect to the given metric, then this equation can be written as The Killing vectors of are the solutions to this equation with The set of all homothety vector fields forms a Lie algebra of vector fields of dimension at most 1 greater than the dimension of the Lie algebra of Killing vector fields. The Killing vector fields are an ideal in the Lie algebra of homotheties.
 • The command HomothetyVectors generates the defining system of 1st order PDE for a homothety vector field and uses pdsolve to find the solutions to these PDE.
 • The command HomothetyVectors returns a sequence of two lists. The first list contains the homothety vector and the second the Killing vectors. If there are no genuine homothety vector fields then the first list is empty.
 • The keyword argument coefficientvariables  allows the user to specify the coefficient functions in the homothety $X$as functions of the variables  .
 • The exact form of the homothety vector $X$can be specified with the keyword argument ansatz  . For example, if the coordinates on the underlying manifold are and  are defined vectors, then one may solve for homothety vectors of the form. In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns
 • When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations Here is a list of the auxiliary equations to be added to the homothety equations.
 • If the metric depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameterswhere is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the homothety vectors changes, are calculated.
 • With keyword argument output = the defining partial differential equations for the homothety vectors are returned. The option output = returns the general homothety vector in terms of a number of arbitrary constants ${\mathrm{_C}}_{1}$, ...  . The option output = returns a list of vectors which form a basis for the solution space. The default value of this keyword argument is output = $"list".$
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form HomothetyVectors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-HomothetyVectors(...).

Examples

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

Example 1.

We find the homotheties for the metric $g$, defined on a 4-dimensional manifold.

 > $\mathrm{DGsetup}\left(\left[u,v,x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $g≔\mathrm{evalDG}\left(\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}-\mathrm{exp}\left(u\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\mathrm{exp}\left(u\right)\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${g}{:=}{\mathrm{du}}{}{\mathrm{du}}{-}\frac{{{ⅇ}}^{{u}}}{{2}}{}{\mathrm{du}}{}{\mathrm{dv}}{-}\frac{{{ⅇ}}^{{u}}}{{2}}{}{\mathrm{dv}}{}{\mathrm{du}}{+}{{ⅇ}}^{{u}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.2)
 M > $H,K≔\mathrm{HomothetyVectors}\left(g\right)$
 ${H}{,}{K}{:=}\left[{2}{}{\mathrm{D_u}}{+}{2}{}{{ⅇ}}^{{-}{u}}{}{\mathrm{D_v}}{+}{y}{}{\mathrm{D_y}}\right]{,}\left[{\mathrm{D_y}}{,}{2}{}{y}{}{\mathrm{D_v}}{+}{{ⅇ}}^{{u}}{}{\mathrm{D_y}}{,}{-}{2}{}{\mathrm{D_u}}{+}{2}{}{v}{}{\mathrm{D_v}}{+}{x}{}{\mathrm{D_x}}{,}{\mathrm{D_x}}{,}{2}{}{x}{}{\mathrm{D_v}}{+}{u}{}{\mathrm{D_x}}{,}{-}{2}{}{\mathrm{D_v}}\right]$ (2.3)

We can check this result by calculating the Lie derivative of the metric with respect to these vector fields (see LieDerivative). We see that the vector field H[1] is a homothety with

 M > $\mathrm{evalDG}\left(\mathrm{LieDerivative}\left(H\left[1\right],g\right)-2g\right)$
 ${0}{}{\mathrm{du}}{}{\mathrm{du}}$ (2.4)
 M > $\mathrm{LieDerivative}\left(K,g\right)$
 $\left[{0}{}{\mathrm{du}}{}{\mathrm{du}}{,}{0}{}{\mathrm{du}}{}{\mathrm{du}}{,}{0}{}{\mathrm{du}}{}{\mathrm{du}}{,}{0}{}{\mathrm{du}}{}{\mathrm{du}}{,}{0}{}{\mathrm{du}}{}{\mathrm{du}}{,}{0}{}{\mathrm{du}}{}{\mathrm{du}}\right]$ (2.5)

We can use the LieAlgebraData command in the LieAlgebras package to calculate the structure equations for the Lie algebra $\mathrm{Γ}$ of homothety vectors.

 M > $\mathrm{Gamma}≔\left[\mathrm{op}\left(K\right),\mathrm{op}\left(H\right)\right]$
 ${\mathrm{Γ}}{:=}\left[{\mathrm{D_y}}{,}{y}{}{\mathrm{D_v}}{+}{{ⅇ}}^{{u}}{}{\mathrm{D_y}}{,}{\mathrm{D_x}}{,}{x}{}{\mathrm{D_v}}{+}{u}{}{\mathrm{D_x}}{,}{-}{2}{}{\mathrm{D_u}}{+}{2}{}{v}{}{\mathrm{D_v}}{+}{x}{}{\mathrm{D_x}}{,}{-}{\mathrm{D_v}}{,}{2}{}{\mathrm{D_u}}{+}{{ⅇ}}^{{-}{u}}{}{\mathrm{D_v}}{+}{y}{}{\mathrm{D_y}}\right]$ (2.6)
 M > $\mathrm{LieAlgebras}:-\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e3}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e6}}\right]$ (2.7)

This output shows, for example, that the Lie bracket of the 1st and 7th vector fields in $\mathrm{Γ}$ is the 1st vector field.

 M > $\mathrm{LieBracket}\left(\mathrm{Gamma}\left[1\right],\mathrm{Gamma}\left[7\right]\right)$
 ${\mathrm{D_y}}$ (2.8)

Example 2.

We look for homotheties of the metric $g$, with the form specified by the vector ${X}_{}$.

 M > $X≔\mathrm{evalDG}\left(\mathrm{D_u}+a\left(u,v\right)\mathrm{D_v}+b\left(u,v\right)y\mathrm{D_y}\right)$
 ${X}{:=}{\mathrm{D_u}}{+}{a}{}\left({u}{,}{v}\right){}{\mathrm{D_v}}{+}{b}{}\left({u}{,}{v}\right){}{y}{}{\mathrm{D_y}}$ (2.9)
 M > $\mathrm{HomothetyVectors}\left(g,\mathrm{ansatz}=X,\mathrm{unknowns}=\left[a\left(u,v\right),b\left(u,v\right)\right]\right)$
 $\left[{\mathrm{D_u}}{+}\left(\frac{{1}}{{2}}{}{{ⅇ}}^{{-}{u}}{+}{1}\right){}{\mathrm{D_v}}{+}\frac{{1}}{{2}}{}{y}{}{\mathrm{D_y}}\right]{,}\left[{}\right]$ (2.10)

Example 3.

We calculate the general homothety vector depending upon 6 arbitrary constants.

 > $\mathrm{HomothetyVectors}\left(g,\mathrm{output}="general"\right)$
 $\left({2}{}{\mathrm{_C3}}{-}{2}{}{\mathrm{_C6}}\right){}{\mathrm{D_u}}{+}\left({{ⅇ}}^{{-}{u}}{}{\mathrm{_C3}}{+}{2}{}{\mathrm{_C6}}{}{v}{+}{\mathrm{_C5}}{}{x}{+}{\mathrm{_C2}}{}{y}{-}{\mathrm{_C7}}\right){}{\mathrm{D_v}}{+}\left({\mathrm{_C6}}{}{x}{+}{\mathrm{_C4}}{+}{\mathrm{_C5}}{}{u}\right){}{\mathrm{D_x}}{+}\left({\mathrm{_C3}}{}{y}{+}{\mathrm{_C1}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{u}}\right){}{\mathrm{D_y}}$ (2.11)

Example 4.

We calculate the homotheties for a metric $g,$which depends upon a parameter There is a true homothety vector only when .

 M > $g≔\mathrm{evalDG}\left(\left({x}^{2}+{y}^{2}+bx\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}+\mathrm{dv}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${g}{:=}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{b}{}{x}\right){}{\mathrm{du}}{}{\mathrm{du}}{+}{\mathrm{dv}}{}{\mathrm{dv}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.12)
 M > $\mathrm{HomothetyVectors}\left(g,\mathrm{parameters}=\left\{b\right\}\right)$
 $\left[\left[{v}{}{\mathrm{D_v}}{+}{x}{}{\mathrm{D_x}}{+}{y}{}{\mathrm{D_y}}\right]{,}\left[{\mathrm{arctan}}{}\left(\frac{{x}}{{y}}\right){}{\mathrm{D_u}}{-}{u}{}{y}{}{\mathrm{D_x}}{+}{u}{}{x}{}{\mathrm{D_y}}{,}{-}{y}{}{\mathrm{D_x}}{+}{x}{}{\mathrm{D_y}}{,}{\mathrm{D_v}}{,}{\mathrm{D_u}}\right]\right]{,}\left[\left[{}\right]{,}\left[{-}{2}{}{y}{}{\mathrm{D_x}}{+}\left({2}{}{x}{+}{b}\right){}{\mathrm{D_y}}{,}{\mathrm{D_v}}{,}{\mathrm{D_u}}\right]\right]{,}\left[\left\{{b}{=}{0}\right\}{,}\left\{{b}{=}{b}\right\}\right]$ (2.13)