Metrics with Exceptional Holonomy

1987/11/01 by Robert L. Bryant · 583 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research #Holonomy #Mathematics #Pure mathematics #Algebra over a field

paper · doi:10.2307/1971360

published in Annals of Mathematics 126(3), 525 (Princeton University)

openalex publication_date 1987/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/12

Abstract

It is proved that there exist metrics with G2 and Spin(7), thus settling the remaining cases in Berger's list of possible groups. We first reformulate the holonomy H condition as a set of differential equations for an associated H-structure on a given manifold. We collect the needed algebraic facts about G2 and Spin(7). We then apply the machinery of over-determined partial differential equations (in the form of the Cartan-Kahler theorem) to prove the existence of solutions whose is G2 or Spin(7). We also provide explicit examples and some information about the generality of the space of

Citations

• Some regularity theorems in riemannian geometry
• Calibrated geometries

Cited by

• On vector bundle manifolds with spherically symmetric metrics
• Perspectives on geometric analysis
• Connections with Irreducible Holonomy Representations
• On the existence of infinite series of exotic holonomies
• On complete quaternionic-Kähler manifolds
• Skew-symmetric complex matrices, pure spinors, the twistor space of the conformal 2n-sphere, and the Fano variety of linear n-folds of a non-singular complex quadric hypersurface in ℙ²ⁿ⁺¹
• The fundamental form of almost-quaternionic Hermitian manifolds
• Introduction to \mathrm G₂ Geometry
• On G_{\mathbf 2}-Structures, Special Metrics and Related Flows
• Special Types of Locally Conformal Closed G2-Structures
• Hyperholomorphic connections on coherent sheaves and stability
• Curvature decomposition of G2-manifolds
• Jacobi relations on naturally reductive spaces
• SU(3)-structures on submanifolds of aSpin(7)-manifold
• Generalised G 2–Manifolds
• Spin(7)-manifolds and symmetric Yang–Mills instantons
• E7, Wirtinger inequalities, Cayley 4-form, and homotopy
• G2 (mathematics) [wikipedia]
• Holonomy [wikipedia]