Existence of compatible contact structures on $G_2$-manifolds

Abstract

Abstract. In this paper, we show the existence of (co-oriented) contact structures on certain classes of G 2 -manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G 2 -structure) admits an almost contact structure. We also construct explicit almost contact metric structures on manifolds with G 2 -structures.

Coclosed G2-structures inducing nilsolitons

“…α 4 α 7 )e 3567 + (2aα 1 − α 2 α 7 + α 3 α 5 − α 4 α 6 )e 1347 + (a 2 − |α| 2 )e 1256 + (a 2 − |α| 2 )e 2357 + (2aα2 + α 1 α 7 − α 3 α 6 − α 4 α 5 )e 2347 + (2aα 2 + α 1 α 7 + α 3 α 6 + α 4 α 5 )e 2567 + (a 2 − |α| 2 )e 1367 + (2aα 4 − α 1 α 6 − α 2 α 5 + α 3 α 7 )e 4567 + (a 2 − |α| 2 )e 1457 + (2aα 1 − α 2 α 7 − α 3 α 5 + α 4 α 6 )e 1567 + (−a 2 + |α| 2 )e 2467 + (a 2 − |α| 2 )e 1234 + aα 1 − α 2 α 7 − α 3 α 5 − α 4 α 6 )e 1236 + (2 − aα 1 − α 2 α 7 + α 3 α 5 + α 4 α 6 )e 1246 + aα 2 + α 1 α 7 + α 3 α 6 − α 4 α 5 )e 1235 + (2aα 3 − α 1 α 5 + α 2 α 6 − α 4 α 7 )e 1237 + (2aα 2 − α 1 α 7 + α 3 α 6 − α 4 α 5 e 1246 + (2aα 4 + α 1 α 6 + α 2 α 5 + α 3 α 7 )e 1247 + (2aα 5 + α 1 α 3 − α 2 α 4 − α 6 α 7 )e 1257 + (2aα 6 − α 1 α 4 − α 2 α 3 + α 5α7 )e 1267 + (2aα 4 + α 1 α 6 − α 2 α 5 − α 3 α 7 e 1345 + (2aα 3 − α 1 α 5 − α 2 α 6 + α 4 α 7 )e 1346 + (−2aα 6 + α 1 α 4 − α 2 α 3 + α 5 α 7 )e 1356 (−2aα 7 − α 1 α 2 − α 3 α 4 − α 5 α 6 )e 1357 + (−2aα 5 − α 1 α 3 − α 2 α 4 − α 6 α 7 )e 1456 + (2aα 7 + α 1 α 2 − α 3 α 4 − α 5 α 6 )e 1467 + (2aα 3 + α 1 α 5 + α 2 α 6 + α 4 α 7 )e 2345 + (−2aα 4 + α 1 α 6 − α 2 α 5 + α 3 α 7 )e 2346 + (−2aα 5 + α 1 α 3 + α 2 α 4 − α 6 α 7 )e 2356 + (2aα 7 − α 1 α 2 + α 3 α 4 − α 5 α 6 )e 2367 + (2aα 6 + α 1 α 4 − α 2 α 3 − α 5 α 7 )e 2456 + (2aα 7 − α 1 α 2 − α 3 α 4 + α 5 α 6 )e 2457 + (2aα 5 − α 1 α 3 + α 2 α 4 − α 6 α 7 )e 3457 + (2aα 6 + α 1 α 4 + α 2 α 3 + α 5 α 7 )e 3467 . Therefore d( * ϕ) = −(a 2 − |α| 2 ) √ 2e 12357 − (2aα 4 − α 1 α α 6 α 7 )e 12345 + − (2aα 3 + α 1 α 5 − α 2 α 6 − α 4 α 7 ) √ 2e 12367 + (a 2 − |α| 2 ) α 5 α 6 )e 12347 + (2aα 6 √ 2 + √ 2α 1 α 4 + √ 2α 2 α 3 + √ 2α 5 α 7 )e 23457 + (−2aα 3 − α 1 α 5 + α 2 α 6 + α 4 α 7 )e 13457 .…”

Conformally parallel Spin(7) structures on solvmanifolds

“…Finally, we mentioned different directions on G 2 and Spin(7) manifolds related to the geometric structures on these spaces. Starting from certain classes of G 2 -manifolds Y , conformally parallel Spin(7) metrics on Riemannian products associated to these manifolds with some special geometric properties should be studied [2,5,33]. All of them give also new research areas related to these exceptional geometries in dimensions d = 7 and d = 8 .…”

“…Many authors have studied special classes of G 2 structures; see, for instance [5,15,23]. Before concentrating on a particular situation, recall that in general the exterior derivatives can be expressed as…”

Almost contact metric structures induced by $G_2$ structures

“…) is an almost contact metric structure on M [4,10]. Throughout this study, (ϕ, ξ, η, g) will denote the almost contact metric structure (a.c.m.s.)…”

“…Arikan et al proved the existence of almost contact metric structures on manifolds with G 2 structures [4]. Todd studied almost contact metric structures on manifolds with parallel G 2 structures [12].…”