Special metrics and scales in parabolic geometry

Abstract

Given a parabolic geometry, it is sometimes possible to find special metrics characterised by some invariant conditions. In conformal geometry, for example, one asks for an Einstein metric in the conformal class. Einstein metrics have the special property that their geodesics are distinguished, as unparameterised curves, in the sense of parabolic geometry. This property characterises the Einstein metrics. In this article, we initiate a study of corresponding phenomena for other parabolic geometries, in particular for the hypersurface CR and contact Legendrean cases.

Appendix: on the equation for conformal circles

Of course, as we saw in the flat model, the reason that distinguished curves in conformal geometry are called conformal circles is that they are modelled on circles in ${\mathbb {R}}^n$. As mentioned in (8), the Lie algebra ${\mathfrak {g}}$, to be used in the Doubrov–Žádník [15] characterisation of conformal circles, is the algebra of conformal Killing fields

$$\begin{aligned} \textstyle (X^b-F^b{}_cx^c+\lambda x^b-Y_cx^cx^b+\frac{1}{2}x_cx^cY^b) \,\partial /\partial x^b \end{aligned}$$

for constant tensors $X^b,F^b{}_c,\lambda ,Y_b$ with $F_{bc}$ skew. The circles through $0\in {\mathbb {R}}^n$ may be parameterised by a pair of vectors

$$\begin{aligned} U^a \text{ of } \text{ unit } \text{ length },\qquad C^a \text{ such } \text{ that } U^aC_a=0. \end{aligned}$$

(31)

Indeed, it is readily verified that

$$\begin{aligned} {\mathbb {R}}\ni t\mapsto \frac{2}{1+t^2C^aC_a}(tU^b+t^2C^b) \end{aligned}$$

(32)

is the circle with velocity $U^a$ and acceleration $C^a$ at the origin, where we are allowing straight lines as ‘circles’ with $C^a=0$.

Lemma 7

The symmetry algebra of the circle (32) is specified by the linear constraints

$$\begin{aligned} X^b=fU^b\qquad F^b{}_cU^c=-fC^b\qquad Y^b=hU^b+\lambda C^b+F^b{}_cC^c \end{aligned}$$

(33)

where f, $\lambda$, and h are arbitrary.

Proof

An elementary verification. $\square$

Note, in particular, that since the symmetry algebra of a circle is non-trivial (of dimension $\big ((n-1)(n-2)/2\big )+3$), circles are homogeneous. It is well known that circles are preserved by conformal transformations. This is why they are suitable model curves in conformal geometry. We also take the opportunity to note that the moduli space of conformal circles in $S^n$ has dimension given by

$$\begin{aligned} \begin{array}{ccccc}\dim {\mathrm {SO}}(n+1,1)&{}-&{} \dim {\mathrm {Sym}}\,\text{ circle }\\ \Vert &{}&{}\Vert \\ (n+2)(n+1)/2&{}-&{}\big ((n-1)(n-2)/2\big )+3&{}=&{}3(n-1), \end{array} \end{aligned}$$

as observed geometrically in Introduction.

To unpack the Doubrov–Žádník characterisation, we shall use the conformal tractor bundle and its connection given, in the presence of a metric, by (10). We shall also need the general form of a tractor endomorphism as in (11) and the tractor directional derivative along $\gamma$ as in (14). We conclude that $\gamma \hookrightarrow M$ is a conformal circle if and only if we can find a subbundle ${\mathcal {S}}$ of the endomorphisms of ${\mathcal {T}}$ along $\gamma$, having the form (11) and such that

• $X^a$ is tangent to $\gamma$,

• fibrewise, the subbundle ${\mathcal {S}}$ has the form (33),

• ${\mathcal {S}}$ is preserved by the tractor connection $\partial \equiv U^a\nabla _a$ along $\gamma$.

To proceed, it is useful to reformulate the conditions (33) as follows.

Lemma 8

In order that an endomorphism (11) satisfy (33), for some fields $U^a$ and $C^a$ satisfying (31), it is necessary and sufficient that

$$\begin{aligned} \begin{array}{c} \Phi \left[ \begin{array}{c}0\\ 0\\ 1\end{array}\right] =\left[ \begin{array}{c} 0\\ -fU_b\\ \lambda \end{array}\right] \qquad \Phi \left[ \begin{array}{c}0\\ U_b\\ 0\end{array}\right] =\left[ \begin{array}{c} f\\ -fC_b\\ -h-fC^bC_b\end{array}\right] \\ [-5pt] \\ \Phi \left[ \begin{array}{c} 1\\ -C_b\\ 0\end{array}\right] =\left[ \begin{array}{c} -\lambda \\ hU_b+\lambda C_b\\ \lambda C^bC_b\end{array}\right] . \end{array}\end{aligned}$$

(34)

Proof

If (33) are satisfied, then it is straightforward to verify that all equations of (34) hold. Conversely, notice that the first two equations from (34) determine f, $U^a$, $\lambda$, $C^a$, and h. The first two constraints from (33) are manifest. If we now substitute in (11), then we find that

$$\begin{aligned} \Phi \left[ \begin{array}{c} 1\\ 0\\ 0\end{array}\right] =\left[ \begin{array}{c} -\lambda \\ Y_b\\ 0\end{array}\right] \quad \text{ and }\quad \Phi \left[ \begin{array}{c}0\\ C_b\\ 0\end{array}\right] =\left[ \begin{array}{c} 0\\ F_{bc}C^c\\ -Y_bC^b\end{array}\right] . \end{aligned}$$

From the second component of the last equation of (34) we conclude that $Y_b=hU_b+\lambda C_b+F_{bc}C^c$, which is the final constraint from (33). $\square$

In fact, three of the conditions in (34) are automatic as follows.

Lemma 9

In order that an endomorphism (11) satisfy (33), for some fields $U^a$ and $C^a$ satisfying (31), it is necessary and sufficient that

$$\begin{aligned} \Phi \!\left[ \begin{array}{c}\!0\!\\ \!0\!\\ \!1\!\end{array}\right] \!\!=\!\!\left[ \begin{array}{c} 0\\ \!\!-fU_b\!\!\\ \lambda \end{array}\right] \Phi \!\left[ \begin{array}{c}0\\ \!U_b\!\\ 0\end{array}\right] \!\!=\!\!\left[ \begin{array}{c} f\\ \!\!-fC_b\!\!\\ *\end{array}\right] \Phi \!\left[ \begin{array}{c} 1\\ \!\!-C_b\!\!\\ 0\end{array}\right] \!\!=\!\!\left[ \begin{array}{c} *\\ \!\!hU_b\!+\!\lambda C_b\!\!\\ *\end{array}\right] \!\!. \end{aligned}$$

(35)

Proof

In proving Lemma 8, we never used the starred components. $\square$

We aim to interpret Theorem 2 as a system of ordinary differential equations on the velocity $U^a$ and acceleration $C^a$ of $\gamma$. Since $X^a$ should be tangent to $\gamma$ we may write $X^b=fU^b$. This is the first constraint from (33) and, of course, our conventions have been chosen for this to be the case. Our next observation justifies $C^a$ as our choice of notation for the acceleration of both $\gamma$ and the corresponding model circle in ${\mathbb {R}}^n$.

Lemma 10

In order to have a subbundle ${\mathcal {S}}$ of ${\mathcal {A}}|_\gamma$, constrained by (33) and being preserved by the tractor directional derivative $\partial$ along $\gamma$, it is necessary that the field $C^b$ in (33) be the acceleration $\partial U^b$ of $\gamma$ (defined with respect to our choice of metric $g_{ab}$).

Proof

According to Lemma 8, the equations (33) are equivalent to (34) and, from (10), the tractor connection $\partial =U^a\nabla _a$ along $\gamma$ is given by

$$\begin{aligned} \partial \left[ \begin{array}{c}\sigma \\ \mu _b\\ \rho \end{array}\right] = \left[ \begin{array}{c}\partial \sigma -U^a\mu _a\\ \partial \mu _b+U_b\rho +U^a\mathrm {P}_{ab}\sigma \\ \partial \rho -U^a\mathrm {P}_a{}^b\mu _b\end{array}\right] . \end{aligned}$$

(36)

The Leibniz rule now calculates the effect of $\partial \Phi$, and, in particular, we find from (34) that

$$\begin{aligned} (\partial \Phi )\!\left[ \begin{array}{c}0\\ 0\\ 1\end{array}\right] \!=\!\partial \!\left[ \begin{array}{c} 0\\ \!\!-fU_b\!\\ \lambda \end{array}\right] - \Phi \partial \! \left[ \begin{array}{c}0\\ 0\\ 1\end{array}\right] \!=\!\left[ \begin{array}{c}0\\ \!\!-(\partial f-\lambda )U_b-f(\partial U_b-C_b)\!\\ \!\partial \lambda \!+\!h\!+\!f(C^aC_a\!+\!\mathrm {P}_{ab}U^aU^b) \end{array}\right] \!\!. \end{aligned}$$

Comparison with (34) shows that $C_b=\partial U_b$, as required. $\square$

For a full interpretation of Theorem 2 in conformal geometry, we are obliged to investigate what it means for the tractor directional derivative $\partial$ to preserve all the equations from (34). Whilst certainly possible, some equations are automatic and, according to Lemma 9, it suffices to start with an endomorphism $\Phi$ satisfying (34) and investigate what it means for $\partial \Phi$ to satisfy (35). This investigation was begun in the proof of Lemma 10 above and next we should consider

$$\begin{aligned} (\partial \Phi )\left[ \begin{array}{c}0\\ U_b\\ 0\end{array}\right] =\partial \Big (\Phi \left[ \begin{array}{c}0\\ U_b\\ 0\end{array}\right] \Big ) -\Phi \partial \left[ \begin{array}{c}0\\ U_b\\ 0\end{array}\right] \!, \end{aligned}$$

the right-hand side of which may be computed from (34) and (36). Indeed, we find

$$\begin{aligned} \begin{array}{lll} \partial \Big (\Phi \left[ \begin{array}{c}0\\ U_b\\ 0\end{array}\right] \Big ) &{}=&{}\partial \left[ \begin{array}{c} f\\ -fC_b\\ -h-fC^bC_b\end{array}\right] \\ [-5pt] \\ &{}=&{}\left[ \begin{array}{c}\partial f\\ -\partial (fC_b)-hU_b-fC^aC_aU_b+fU^a\mathrm {P}_{ab}\\ -\partial (h+fC^aC_a)+f\mathrm {P}_{ab}U^aC^b\end{array}\right] \end{array} \end{aligned}$$

and

$$\begin{aligned} \Phi \partial \left[ \begin{array}{c}0\\ U_b\\ 0\end{array}\right] =\Phi \left[ \begin{array}{c}-1\\ C_b\\ -\mathrm {P}_{ab}U^aU^b\end{array}\right] =\left[ \begin{array}{c} \lambda \\ (f\mathrm {P}_{ac}U^aU^c-h)U_b-\lambda C_b\\ -\lambda (C^bC_b+\mathrm {P}_{bc}U^bU^c)\end{array}\right] \end{aligned}$$

so

$$\begin{aligned} (\partial \Phi )\left[ \begin{array}{c}0\\ U_b\\ 0\end{array}\right] =\left[ \begin{array}{c}\partial f-\lambda \\ -(\partial f-\lambda )C_b -fE_b\\ *\end{array}\right] , \end{aligned}$$

where $E_b\equiv \partial C_b-\mathrm {P}_{bc}U^c+(C^aC_a+\mathrm {P}_{ac}U^aU^c)U_b$ and, for our present purposes, it does not matter what is the last entry. We conclude that ${\mathcal {S}}$ is preserved along $\gamma$ in accordance with Theorem 2, if and only if $E_b$ is identically zero, which is the conformal circles Eq. (5).

Finally, to complete our investigation of (35), we should compute

$$\begin{aligned} (\partial \Phi )\left[ \begin{array}{c}1\\ -C_b\\ 0\end{array}\right] =\partial \Big (\Phi \left[ \begin{array}{c}1\\ -C_b\\ 0\end{array}\right] \Big ) -\Phi \partial \left[ \begin{array}{c}1\\ -C_b\\ 0\end{array}\right] \!. \end{aligned}$$

Well, from (34), we find

$$\begin{aligned} \begin{array}{lll} \partial \Big (\Phi \left[ \begin{array}{c}1\\ -C_b\\ 0\end{array}\right] \Big ) &{}=&{}\partial \left[ \begin{array}{c} -\lambda \\ hU_b+\lambda C_b\\ \lambda C^aC_a\end{array}\right] \\ [-5pt] \\ &{}=&{}\left[ \begin{array}{c} -\partial \lambda -h\\ \partial (hU_b+\lambda C_b)+\lambda C^aC_aU_b-\lambda \mathrm {P}_{ab}U^a\\ *\end{array}\right] \end{array} \end{aligned}$$

and

$$\begin{aligned} \Phi \partial \left[ \begin{array}{c}1\\ -C_b\\ 0\end{array}\right] =\Phi \left[ \begin{array}{c} 0\\ -\partial C_b+U^a\mathrm {P}_{ab}\\ U^a\mathrm {P}_a{}^bC_b\end{array}\right] =\Phi \left[ \begin{array}{c} 0\\ (C^aC_a+\mathrm {P}_{ac}U^aU^c)U_b\\ \mathrm {P}_{ab}U^aC^b\end{array}\right] , \end{aligned}$$

where we have just used (5) to rewrite $\partial C_b-\mathrm {P}_{ab}U^a$. From here,

$$\begin{aligned} \Phi \partial \left[ \begin{array}{c}1\\ -C_b\\ 0\end{array}\right] =\left[ \begin{array}{c} f(C^aC_a+\mathrm {P}_{ac}U^aU^c)\\ -f\mathrm {P}_{ac}U^aC^cU_b-f(C^aC_a+\mathrm {P}_{ac}U^aU^c)C_b\\ * \end{array}\right] . \end{aligned}$$

Also note that (5) allows us to write

$$\begin{aligned} \partial (hU_b+\lambda C_b)+\lambda C^aC_aU_b-\lambda \mathrm {P}_{ab}U^a =(\partial h-\lambda \mathrm {P}_{ac}U^aU^c)U_b+(\partial \lambda +h)C_b \end{aligned}$$

and so, finally,

$$\begin{aligned} (\partial \Phi )\left[ \begin{array}{c}1\\ -C_b\\ 0\end{array}\right] =\left[ \begin{array}{c} -\partial \lambda -h-f(C^aC_a+\mathrm {P}_{ac}U^aU^c)\\[2pt] \begin{array}{l}(\partial h-\lambda \mathrm {P}_{ac}U^aU^c+f\mathrm {P}_{ac}U^aC^c)U_b\\ [-1pt] \qquad {}+(\partial \lambda +h+f(C^aC_a+\mathrm {P}_{ac}U^aU^c))C_b\end{array}\\[8pt] *\end{array}\right] , \end{aligned}$$

in accordance with (35). We have proved the following.

Theorem 5

Let M be a Riemannian manifold. An unparameterised curve $\gamma \hookrightarrow M$ is distinguished in the sense of Theorem 2, modelled on the circles in ${\mathbb {R}}^n$, if and only if (5) holds along $\gamma$.