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We show that $3$-Sasaki structures admit a natural description in terms of projective differential geometry. This description provides a concrete link between $3$-Sasaki structures and several other geometries and constructions via a single unifying picture. First we establish that a $3$-Sasaki structure may be understood as a projective structure equipped with a certain holonomy reduction to the (possibly indefinite) unitary quaternionic group $\textrm{Sp}(p,q)$, namely a parallel hyperk\"ahler structure on the projective tractor bundle satisfying a particular genericity condition. For the converse, where one begins with a general parallel hyperk\"ahler structure on the projective tractor bundle, the genericity condition is not automatic. Indeed we prove that generically such a reduction decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) $3$-Sasaki structures and a closed separating hypersurface at infinity with respect to the $3$-Sasaki metrics. Moreover, it is shown that the latter hypersurface inherits a Biquard-Fefferman conformal structure, which thus (locally) fibres over a quaternionic contact structure, and which in turn compactifies the natural quaternionic K\"ahler quotients of the $3$-Sasaki structures on the open manifolds. As an application we describe the projective compactification of (suitably) complete, non-compact (indefinite) $3$-Sasaki manifolds and recover Biquard's notion of asymptotically hyperbolic quaternionic K\"ahler metrics., Comment: 26 pages

We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature $(1,1)$ on a contact 3-manifold. This approach is significantly simpler than the standard method of solving the full equivalence problem for path geometry. We then use these equations to give a characterization of projective path geometries in terms of their chains (the chains projected to the surface coincide with the paths) and study the chains of four examples of homogeneous path geometries. In one of these examples (horocycles in the hyperbolic planes) the projected chains are bicircular quartics., Comment: 35 pages, 5 figures (corrected a spelling error in an author's name)

We classify multiply transitive homogeneous real (2,3,5) distributions up to local diffeomorphism equivalence.

We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete non-compact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the K\"ahler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail., Comment: 55 pages; minor modifications: references updated, citations added and typos corrected. To appear in Dissertationes Mathematicae

For the geometry of oriented $(2, 3, 5)$ distributions $(M, {\mathbf D})$, which correspond to regular, normal parabolic geometries of type $(\mathrm{G}_2, P)$ for a particular parabolic subgroup $P < \mathrm{G}_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator $\Theta_0$ associated to the $7$-dimensional irreducible representation of $\mathrm{G}_2$. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator are automatically normal, yielding a geometric interpretation of $\ker \Theta_0$: For any $(M, {\mathbf D})$, this kernel consists precisely of the almost Einstein scales of the Nurowski conformal structure on $M$ that ${\mathbf D}$ determines. We apply our formula for $\Theta_0$ (1) to recover efficiently some known solutions, (2) to construct a distribution with root type $[3, 1]$ with a nonzero solution, and (3) to show efficiently that the conformal holonomy of a particular $(2, 3, 5)$ conformal structure is equal to $\mathrm{G}_2$., Comment: Move proof of Theorem 1 to the introduction. Removed some extraneous content from material about Weyl connections. Restructured section hierarchy. Made many adjustments for clarity. Adjusted the explicit adjoint representation (see appendix). 16 pages

We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures $\mathbf{c}$ that are induced by at least two distinct oriented (2,3,5) distributions; in this case there is a 1-parameter family of such distributions that induce $\mathbf{c}$. Second, they are characterized by the existence of a holonomy reduction to $SU(1,2)$, $SL(3,{\mathbb R})$, or a particular semidirect product $SL(2,{\mathbb R})\ltimes Q_+$, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2,3,5) distributions and many other geometries - several classical geometries among them - including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; K\"ahler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional K\"ahler-Einstein or para-K\"ahler-Einstein structure a family of (2,3,5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2,3,5) conformal structures for which the Einstein constant is positive and negative.

In his 1910 "Five Variables" paper, Cartan solved the equivalence problem for the geometry of $(2, 3, 5)$ distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type $\textrm{G}_2$. He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least $6$ (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution $\bf E$. We produce a closed form for the Fefferman-Graham ambient metric $\widetilde{g}_{\bf E}$ of the conformal class induced by (a real form of) $\bf E$, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of $\widetilde{g}_{\bf E}$ is the exceptional group $\textrm{G}_2^*$ and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for $\widetilde{g}_{\bf E}$., Comment: 16 pages. The proof of Proposition 3 has been simplified (the original is now sketched in Remark 4). Section 4 has been significantly revised for clarity. The proofs of Propositions 7 and 11 are now more explicit. Remark 12, which contains a proof that the main example given here is not among thethose given in arXiv:1501.00852, was added. Some inessential material was cut for brevity

We show that any dimension $6$ nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel $7$-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly K\"ahler manifold, if the cross product is definite, otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly K\"ahler and nearly para-K\"ahler parts separated by a hypersurface which canonically carries a Cartan $(2,3,5)$-distribution. This hypersurface is a projective infinity for the pseudo-Riemannian geometry elsewhere on the manifold, and we establish how the Cartan distribution can be understood explicitly, and also in terms of conformal geometry, as a limit of the ambient nearly (para\nobreakdash-)K\"ahler structures. Any real-analytic $(2,3,5)$-distribution is seen to arise as such a limit, because we can solve the geometric Dirichlet problem of building a collar structure equipped with the required holonomy-reduced projective structure. A model geometry for these structures is provided by the projectivization of the imaginary (split) octonions. Our approach is to use Cartan/tractor theory to provide a curved version of this geometry, this encodes a curved version of the algebra of imaginary (split) octonions as a flat structure over its projectivization. The perspective is used to establish detailed results concerning the projective compactification of nearly (para\nobreakdash-)K\"ahler manifolds, including how the almost (para\nobreakdash-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there., Comment: 48 pages. Theorem 1.2 corrected and simplified. Outlook section and four of six appendices removed. Minor formatting and expository changes made

Nurowski showed that any generic 2-plane field $D$ on a 5-manifold $M$ determines a natural conformal structure $c_D$ on $M$; these conformal structures are exactly those (on oriented $M$) whose normal conformal holonomy is contained in the (split, real) simple Lie group $G_2$. Graham and Willse showed that for real-analytic $D$ the same holds for the holonomy of the real-analytic Fefferman-Graham ambient metric of $c_D$, and that both holonomy groups are equal to $G_2$ for almost all $D$. We investigate here independently interesting plane fields for which the associated holonomy groups are a proper subset of $G_2$. Cartan solved the local equivalence problem for $2$-plane fields $D$ and constructed the fundamental curvature tensor $A$ for these objects. He furthermore claimed to describe locally all $D$ whose infinitesimal symmetry algebra has rank at least $6$ and gave a local quasi-normal form, depending on a single function of one variable, for those that furthermore satisfy a natural degeneracy condition on $A$, but Doubrov and Govorov recently rediscovered a counterexample to Cartan's claim. We show that for all $D$ given by Cartan's alleged quasi-normal form, the conformal structures $c_D$ induced via Nurowski's construction are almost Einstein, that we can write their ambient metrics explicitly, and that the holonomy groups associated to $c_D$ are always the $5$-dimensional Heisenberg group, which here acts indecomposably but not irreducibly. (Not all of these properties hold, however, for Doubrov and Govorov's counterexample.) We also show that the similar results hold for the related class of $2$-plane fields defined on suitable jet spaces by ordinary differential equations $z'(x) = F(y''(x))$ satisfying a simple genericity condition., Comment: 34 pages. Revised to accommodate a counterexample to a cited classification of Cartan found by Doubrov and Govorov; fixed some minor errors

The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures., Comment: 17 pages

The holonomy of the ambient metrics of Nurowski's conformal structures associated to generic real-analytic 2-plane fields on 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G_2 of the exceptional Lie group, and is equal to G_2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split G_2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G_2 is established by proving an ambient extension theorem for parallel tractors for conformal structures in general signature and dimension, which is expected to be of independent interest. Parallel extension beyond the critical order in even dimensions is considered in certain cases., Comment: 39 pages