TY - JOUR
T1 - Nearly Kahler geometry and (2, 3, 5)-distributions via projective holonomy
AU - Gover, A. Rod
AU - Panai, Roberto
AU - Willse, Travis
PY - 2017
Y1 - 2017
N2 - We show that any dimension-6 nearly Kahler (or nearly para-Kahler) geometry arises as a projective manifold equipped with a G2(∗) holonomy reduction. In the converse direction, we show that if a projective manifold is equipped with a parallel seven-dimensional cross product on its standard tractor bundle, then the manifold is a Riemannian nearly Kahler 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 Kahler and nearly para- Kahler parts separated by a hypersurface that 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-)Kahler 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-)Kahler manifolds, including how the almost (para-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there.
AB - We show that any dimension-6 nearly Kahler (or nearly para-Kahler) geometry arises as a projective manifold equipped with a G2(∗) holonomy reduction. In the converse direction, we show that if a projective manifold is equipped with a parallel seven-dimensional cross product on its standard tractor bundle, then the manifold is a Riemannian nearly Kahler 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 Kahler and nearly para- Kahler parts separated by a hypersurface that 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-)Kahler 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-)Kahler manifolds, including how the almost (para-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there.
KW - Conformal differential geometry
KW - Einstein metrics
KW - G geometry
KW - Holography
KW - Nearly Kahler
KW - Projective differential geometry
UR - http://www.scopus.com/inward/record.url?scp=85034049026&partnerID=8YFLogxK
U2 - 10.1512/iumj.2017.66.6089
DO - 10.1512/iumj.2017.66.6089
M3 - Article
SN - 0022-2518
VL - 66
SP - 1351
EP - 1416
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
IS - 4
ER -