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Aerodynamics of Flying Saucers

Abstract

We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of ‘parabolic geometry’) and we provide examples when this geometry is ‘flat,’ meaning that its symmetries comprise the split form of the exceptional Lie algebra \({\mathfrak {g}}_2\).

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Correspondence to Paweł Nurowski.

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Communicated by P. Chrusciel

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This work was supported by the Simons Foundation Grant 346300 and the Polish Government MNiSW 2015–2019 matching fund. It was written whilst the first author was visiting the Banach Centre at IMPAN in Warsaw for the Simons Semester ‘Symmetry and Geometric Structures’ and during another visit to Warsaw supported by the Polish National Science Centre (NCN) via the POLONEZ Grant 2016/23/P/ST1/04148, which received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665778

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Appendix: Harmonic Curvature of a \(G_2\) Contact Structure

Appendix: Harmonic Curvature of a \(G_2\) Contact Structure

As already mentioned (see [2] for the general theory), the harmonic curvature of a \(G_2\) contact structure is a section of the bundle . This binary septic may be obtained by the Cartan equivalence method.

Specifically, a \(G_2\) contact structure on a five-dimensional manifold C may be specified in terms of an adapted co-frame as follows. Firstly, we choose a 1-form \(\omega ^0\) whose kernel is the contact distribution \(H\subset TC\). Non-degeneracy of the contact distribution says that \(\omega ^0\wedge d\omega ^0\wedge d\omega ^0\not =0\). The \(G_2\) structure is then determined by completing \(\omega ^0\) to a co-frame \(\omega ^0,\omega ^1,\omega ^2,\omega ^3,\omega ^4\) so that

$$\begin{aligned} d\omega ^0=\chi (\omega ^1\wedge \omega ^4-3\omega ^2\wedge \omega ^3) \bmod \omega ^0 \end{aligned}$$
(27)

for some smooth function \(\chi \). Specifically, if \(X_0,X_1,X_2,X_3,X_4\) is the dual frame, then \(H={\mathrm {span}}\{X_1,X_2,X_3,X_4\}\) and the twisted cubic (21) may be given as

$$\begin{aligned} (s,t)\mapsto s^3X_1+s^2tX_2+st^2X_3+t^3X_4, \end{aligned}$$
(28)

compatibility with the Levi form being a consequence of (27). Imposing the structure equations (27) leaves precisely the following freedom in choice of co-frame:

$$\begin{aligned} \left[ \begin{array}{c}{\tilde{\omega }}^0\\ {\tilde{\omega }}^1\\ {\tilde{\omega }}^2\\ {\tilde{\omega }}^3\\ {\tilde{\omega }}^4\end{array}\right] =\left[ \begin{array}{ccccc} t_9&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ t_{10}&{}\quad t_5{}^3&{}\quad 3t_5{}^2t_6&{}\quad 3t_5t_6{}^2&{}\quad t_6{}^3\\ t_{11}&{}\quad t_5{}^2t_7&{}\quad t_5(t_5t_8+2t_6t_7)&{}\quad t_6(2t_5t_8+t_6t_7)&{}\quad t_6{}^2t_8\\ t_{12}&{}\quad t_5t_7{}^2&{}\quad t_7(2t_5t_8+t_6t_7)&{}\quad t_8(t_5t_8+2t_6t_7)&{}\quad t_6t_8{}^2\\ t_{13}&{}\quad t_7{}^3&{}\quad 3t_7{}^2t_8&{}\quad 3t_7t_8{}^2&{}\quad t_8{}^3 \end{array}\right] \, \left[ \begin{array}{c}\omega ^0\\ \omega ^1\\ \omega ^2\\ \omega ^3\\ \omega ^4\end{array}\right] \end{aligned}$$
(29)

for arbitrary functions \(t_5,t_6,t_7,t_8,t_9,t_{10},t_{11},t_{12},t_{13}\) on C subject only to \(t_9(t_5t_8-t_6t_7)\not =0\). (The functions \(t_5,t_6,t_7,t_8\) correspond to

$$\begin{aligned} {(s,t)\mapsto (s,t)} \left( \begin{array}{cc}t_5&{}\quad t_7\\ t_6&{}\quad t_8\end{array}\right) \end{aligned}$$

as a change of parameterisation in (28) whilst \(t_9,t_{10},t_{11},t_{12},t_{13}\) modify the co-frame with multiples of \(\omega ^0\).) To proceed with Cartan’s method of equivalence, we now pass to the bundle \(\widetilde{C}\rightarrow C\) whose sections are frames adapted according to (27). It is a G-principal bundle where G is the 9-dimensional Lie subgroup of \({\mathrm {GL}}(5,{\mathbb {R}})\) given in (29) and comes tautologically equipped with 1-forms \(\theta ^0,\theta ^1,\theta ^2,\theta ^3,\theta ^4\) whose pull-backs along a section are \(\omega ^0,\omega ^1,\omega ^2,\omega ^3,\omega ^4\), the co-frame on C corresponding to that section. Cartan’s aim is to extend this to an invariant co-frame on \(\widetilde{C}\) by making various normalisations. For our purposes we need not take these normalisations too far. For calculation, we choose a co-frame on C adapted according to (27) so that \(\widetilde{C}\) is identified as \(G\times C\) and the forms \(\theta ^0,\theta ^1\,\theta ^2,\theta ^3,\theta ^4\) are given as

$$\begin{aligned} \left[ \begin{array}{c}\theta ^0\\ \theta ^1\\ \theta ^2\\ \theta ^3\\ \theta ^4\end{array}\right] =\left[ \begin{array}{ccccc} t_9&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ t_{10}&{}\quad t_5{}^3&{}\quad 3t_5{}^2t_6&{}\quad 3t_5t_6{}^2&{}\quad t_6{}^3\\ t_{11}&{}\quad t_5{}^2t_7&{}\quad t_5(t_5t_8+2t_6t_7)&{}\quad t_6(2t_5t_8+t_6t_7)&{}\quad t_6{}^2t_8\\ t_{12}&{}\quad t_5t_7{}^2&{}\quad t_7(2t_5t_8+t_6t_7)&{}\quad t_8(t_5t_8+2t_6t_7)&{}\quad t_6t_8{}^2\\ t_{13}&{}\quad t_7{}^3&{}\quad 3t_7{}^2t_8&{}\quad 3t_7t_8{}^2&{}\quad t_8{}^3 \end{array}\right] \, \left[ \begin{array}{c}\omega ^0\\ \omega ^1\\ \omega ^2\\ \omega ^3\\ \omega ^4\end{array}\right] . \end{aligned}$$

Step 0  Normalise the co-frame on C so that

$$\begin{aligned} d\omega ^0=\chi (\omega ^1\wedge \omega ^4-3\omega ^2\wedge \omega ^3). \end{aligned}$$
(30)

This is easily achieved by the freedom

$$\begin{aligned} {\tilde{\omega }}^1=\omega ^1+t_{10}\omega ^0,{\tilde{\omega }}^2=\omega ^1+t_{11}\omega ^0,{\tilde{\omega }}^3=\omega ^1+t_{12}\omega ^0,{\tilde{\omega }}^4=\omega ^1+t_{13}\omega ^0 \end{aligned}$$

and determines the functions \(t_{10},t_{11},t_{12},t_{13}\).

Step 1   Find \(\theta ^5\) such that

$$\begin{aligned} d\theta ^0 =-6\theta ^0\wedge \theta ^5+\theta ^1\wedge \theta ^4-3\theta ^2\wedge \theta ^3. \end{aligned}$$

This may be achieved by setting \(-t_9=\Delta ^3/\chi \), where \(\Delta =t_6t_7-t_5t_8\) and then

$$\begin{aligned} \theta ^5=-\frac{1}{6}\frac{d\chi }{\chi }+\frac{1}{2}\frac{d\Delta }{\Delta } +\frac{\chi }{\Delta ^3}\Big (\frac{t_{13}\theta ^1}{6}-\frac{t_{12}\theta ^2}{2} +\frac{t_{11}\theta ^3}{2}-\frac{t_{10}\theta ^4}{6}\Big )+s\theta ^0 \end{aligned}$$

for some function s.

Step 2   Find \(\theta ^7,\theta ^8,\theta ^9\) such that

$$\begin{aligned} E^1\equiv d\theta ^1-(6\theta ^0\wedge \theta ^9-3\theta ^1\wedge \theta ^5 -3\theta ^1\wedge \theta ^8+3\theta ^2\wedge \theta ^7) \end{aligned}$$

is of the form \(c^1{}_{\mu \nu }\theta ^\mu \wedge \theta ^v\) for \(\mu ,\nu =0,1,2,3,4\). It follows that

$$\begin{aligned} E^1\wedge \theta ^0\wedge \theta ^1\wedge \theta ^2 =c^1{}_{34}\theta ^0\wedge \theta ^1\wedge \theta ^2\wedge \theta ^4\wedge \theta ^5. \end{aligned}$$

It turns out that

$$\begin{aligned} c^1{}_{34}=\frac{t_5{}^7}{\Delta ^5} \Big (\psi _0+\psi _1s+\psi _2s^2+\psi _3s^3+\psi _4s^4+\psi _5s^5+\psi _6s^6 +\psi _7s^7\Big ), \end{aligned}$$

where \(\psi _0,\psi _1,\psi _2,\psi _3,\psi _4,\psi _5,\psi _6,\psi _7\) are functions on C. These are the coefficients of the invariantly defined harmonic curvature. In fact, if

$$\begin{aligned} d\omega ^0=\chi (\omega ^1\wedge \omega ^4-3\omega ^2\wedge \omega ^3) \end{aligned}$$

on C and we write

$$\begin{aligned} d\omega ^i=\sum _{1\le j<k\le 4}a^i{}_{jk}\omega ^j\wedge \omega ^k \bmod \omega ^0,\text{ for } i=1,2,3,4 \end{aligned}$$
(31)

then

$$\begin{aligned} \begin{array}{rcl}\psi _0&{}=&{}a^1{}_{34}\\ \psi _1&{}=&{}-2a^1{}_{24}+3a^2{}_{34}\\ \psi _2&{}=&{}3a^1{}_{14}+a^1{}_{23}-6a^2{}_{24}+3a{}^3{}_{34}\\ \psi _3&{}=&{}-2a^1{}_{13}+9a^2{}_{14}+3a^2{}_{23}-6a^3{}_{24}+a^4{}_{34}\\ \psi _4&{}=&{}a^1{}_{12}-6a^2{}_{13}+9a^3{}_{14}+3a^3{}_{23}-2a^4{}_{24}\\ \psi _5&{}=&{}3a^2{}_{12}-6a^3{}_{13}+3a^4{}_{14}+a^4{}_{23}\\ \psi _6&{}=&{}3a^3{}_{12}-2a^4{}_{13}\\ \psi _7&{}=&{}a^4{}_{12}. \end{array} \end{aligned}$$
(32)

From the general theory of parabolic geometry [2, Theorem 3.1.12] we find the following characterisation of flat \(G_2\) contact structures.

Theorem 5

The local symmetry algebra of the \(G_2\) contact structure specified by a co-frame adapted according to (30) is the split exceptional Lie algebra \(G_2\) if and only if \(\psi _0,\psi _1,\psi _2,\psi _3,\psi _4,\psi _5,\psi _6,\psi _7\) given by (32) all vanish. Moreover, in this case, the manifold C is locally isomorphic to the homogeneous model.

Remark

The formulæ (31) and (32) may alternatively be derived as follows. According to (20), the exterior derivative \(d:\wedge ^1\rightarrow \wedge ^2\) gives rise, via the diagram

to an invariantly defined first order differential operator

(33)

In fact, this is nothing more than the second operator in the Rumin complex [16], which depends only on the contact structure on C. The exterior derivative, on the other hand, may be written as \(\omega _b\mapsto \nabla _{[a}\omega _{b]}\) for any torsion-free connection \(\nabla _a\) on TC. Thus, the Rumin operator (33) may be written with spinor indices [15], adapted to our cause, as

$$\begin{aligned} \omega _{ABC}\longmapsto \nabla _{(AB}{}^H\omega _{CD)H}. \end{aligned}$$
(34)

One may readily check that the formulæ (31) and (32) amount to the stipulation that

$$\begin{aligned} \psi _{ABCDEFG}\pi ^A\pi ^B\pi ^C\pi ^D\pi ^E\pi ^F\pi ^G =\pi ^A\pi ^B\pi ^C\pi ^D\nabla _{AB}{}^H(\pi _C\pi _D\pi _H) \end{aligned}$$
(35)

for all sections \(\pi _A\) of  . Note, by the Leibniz rule

$$\begin{aligned} \begin{array}{rcl} \pi ^A\pi ^B\pi ^C\pi ^D\nabla _{AB}{}^H(f\pi _C\pi _D\pi _H) &{}=&{}f\pi ^A\pi ^B\pi ^C\pi ^D\nabla _{AB}{}^H(\pi _C\pi _D\pi _H)\\ &{}&{}{}+\pi ^A\pi ^B\underbrace{\pi ^C\pi ^D\pi _C}_{=0}\pi _D\pi _H \nabla _{AB}{}^Hf,\end{array} \end{aligned}$$

that the right hand side of (35) is homogeneous of degree 7 over the functions and, therefore, automatically of the form given on the left. It follows that \(\psi _{ABCDEFG}\) is the obstruction to writing (33) as

$$\begin{aligned} \omega _{ABC}\longmapsto {\mathcal {D}}_{(AB}{}^H\omega _{CD)H}, \end{aligned}$$

where \({\mathcal {D}}_{ABC}\) is induced by a connection on    , because the Leibniz rule for such a connection would imply that

$$\begin{aligned} {\mathcal {D}}_{AB}{}^H(\pi _C\pi _D\pi _H) =\pi _C\pi _D{\mathcal {D}}_{AB}{}^H\pi _H +\pi _C\pi _H{\mathcal {D}}_{AB}{}^H\pi _D +\pi _D\pi _H{\mathcal {D}}_{AB}{}^H\pi _C \end{aligned}$$

and the right hand side of (35) would therefore vanish. In other words, the formula (34) depends on \(\nabla _a\) being torsion-free and \(\psi _{ABCDEFG}\) may be seen as some invariant part of the partial torsion of a freely chosen spinor connection \({\mathcal {D}}_a:S\rightarrow \wedge ^1\otimes S\).

More specifically, suppose \({\mathcal {D}}_a:S\rightarrow \wedge ^1\otimes S\) is any connection and define its partial torsion\(T_{ABCD}{}^{EFG}=T_{(ABCD)}{}^{(EFG)}\) according to

$$\begin{aligned} {\mathcal {D}}_{(AB}{}^E{\mathcal {D}}_{CD)E}f =T_{ABCD}{}^{EFG}{\mathcal {D}}_{EFG}f,\forall \text{ smooth } \text{ functions } f. \end{aligned}$$

Changing the connection \({\mathcal {D}}_a\), leads to a change of partial connection according to

$$\begin{aligned} \widehat{\mathcal {D}}_{ABC}\pi _D ={\mathcal {D}}_{ABC}\pi _D-\Gamma _{ABCD}{}^E\pi _E, \text{ where } \Gamma _{ABCD}{}^E=\Gamma _{(ABC)D}{}^E \end{aligned}$$

and, therefore, an induced change on , namely

$$\begin{aligned} \widehat{\mathcal {D}}_{ABC}\omega _{DEF}= {\mathcal {D}}_{ABC}\omega _{DEF}-3\Gamma _{ABC(D}{}^G\omega _{EF)G}. \end{aligned}$$

It follows that

$$\begin{aligned} \widehat{\mathcal {D}}_{AB}{}^E\widehat{\mathcal {D}}_{CDE}f ={\mathcal {D}}_{AB}{}^E{\mathcal {D}}_{CDE}f -3\Gamma _{AB}{}^E{}_{(C}{}^G{\mathcal {D}}_{DE)G}f \end{aligned}$$

and, therefore, that

$$\begin{aligned} \widehat{\mathcal {D}}_{(AB}{}^E\widehat{\mathcal {D}}_{CD)E}f = {\mathcal {D}}_{(AB}{}^E{\mathcal {D}}_{CD)E}f -2\Gamma _{(AB}{}^E{}_C{}^F{\mathcal {D}}_{D)EF}f +\Gamma _{H(AB}{}^{HE}{\mathcal {D}}_{CD)E}f \end{aligned}$$

whence the partial torsion of \({\mathcal {D}}_a\) changes according to

$$\begin{aligned} \widehat{T}_{ABCD}{}^{EFG}=T_{ABCD}{}^{EFG} -2\Gamma _{(AB}{}^{(E}{}_C{}^F\delta _{D)}{}^{G)} +\Gamma _{H(AB}{}^{H(E}\delta _C{}^F\delta _{D)}{}^{G)}. \end{aligned}$$

In particular, the trace-free part of \(T_{ABCD}{}^{EFG}\), equivalently

$$\begin{aligned} \psi _{ABCDEFG}\equiv T_{(ABCDEFG)}, \end{aligned}$$

is an invariant of the \(G_2\) contact structure.

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Eastwood, M., Nurowski, P. Aerodynamics of Flying Saucers. Commun. Math. Phys. 375, 2367–2387 (2020). https://doi.org/10.1007/s00220-019-03622-1

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