7 ManifoldEdit
I can’t frame this topic as political persuasion, but here is a neutral, encyclopedia-style overview of 7-manifolds, with emphasis on geometry, topology, and key constructions.
A 7-manifold is a smooth manifold of dimension seven. In differential geometry it serves as a natural arena for exploring exceptional geometric structures and calibrated submanifolds. A central object of study is the existence of special holonomy, particularly the group G2, which arises as the automorphism group of the octonions. When a 7-manifold admits a torsion-free G2-structure, its Riemannian metric has holonomy contained in G2, making the manifold Ricci-flat and highly constrained topologically. This interplay between analysis, topology, and geometry has made seven-dimensional manifolds a focal point in both pure mathematics and theoretical physics, notably in certain compactifications of higher-dimensional theories.
Geometric structures on 7-manifolds
A G2-structure on a smooth 7-manifold M is given by a differential 3-form φ that is pointwise equivalent to the standard stable 3-form preserved by G2. The form φ determines a Riemannian metric gφ and an orientation. The associated Hodge star operator φ then defines a complementary 4-form *φ. A G2-structure is torsion-free if dφ = 0 and dφ = 0; in this case the Levi-Civita holonomy group of the metric is contained in G2, and, for compact simply connected M, the holonomy is exactly G2 for a full-structure reality.
- The 3-form φ encodes the metric and the orientation, and it serves as a calibrating form in calibrated geometry, yielding distinguished submanifolds.
- The automorphism group G2 is intimately connected to the octonions octonions, and the study of G2-structures often draws on this algebraic background.
- Deformations of torsion-free G2-structures are governed by harmonic 3-forms. On a compact M, the moduli space of torsion-free G2-structures is smooth and its dimension is b^3(M), the third Betti number, reflecting the number of independent deformations.
In addition to the torsion-free case, there are weaker G2-structures (often called nearly parallel or cocalibrated) where only some components of torsion vanish. These produce interesting geometric and topological consequences, but do not yield holonomy exactly equal to G2.
Within this framework, two principal notions of submanifolds emerge:
- Associative submanifolds are 3-dimensional and calibrated by φ, giving minimal-volume representatives in their homology class.
- Coassociative submanifolds are 4-dimensional and calibrated by *φ, again providing minimal-volume representatives.
These submanifolds illuminate the rich calibrated geometry associated with G2-structures and connect to broader questions in minimal surface theory and geometric analysis. See also associative submanifold and coassociative submanifold.
Other standard topics include holonomy theory, the classification program for special holonomy groups (Berger’s classification), and the relationship between holonomy and the curvature tensor. For a general introduction to the concept of holonomy, see holonomy.
Examples and constructions
The round 7-sphere S^7 carries a natural, highly symmetric structure that can be viewed as an almost G2-structure; however, it is not torsion-free. It serves as a classical example illustrating nearly parallel G2-structures, a weaker condition than torsion-free. See S^7.
Complete non-compact G2-manifolds were first constructed explicitly by Bryant and Salamon. These include metrics on total spaces of certain vector bundles over lower-dimensional spheres, providing important model geometries for the local and asymptotic theory of G2-structures. See Bryant–Salamon.
Compact G2-manifolds were constructed by Dominic Joyce by resolving singularities of toroidal orbifolds T^7/Γ and then smoothing them to obtain smooth, compact manifolds with holonomy exactly G2. Joyce’s work established the existence of compact seven-dimensional manifolds with exceptional holonomy and spurred a wide range of further developments. See Joyce, Dominic.
The twisted connected sum construction, developed by Kovalev and refined by others, produces many compact G2-manifolds by gluing together asymptotically cylindrical Calabi–Yau 3-folds times S^1 along their cylindrical ends. This method greatly expanded the catalog of known G2-manifolds and provided a flexible toolkit for exploring moduli and topology. See twisted connected sum.
Non-compact and asymptotically cylindrical or asymptotically conical G2-manifolds arise as limits or local models for compact G2-geometry. They are important for understanding analysis on G2-structures, including the behavior of differential operators and moduli theory near infinity.
In all these constructions, the central object remains the same: a 3-form φ encoding a geometric structure whose torsion properties determine the holonomy and curvature of the induced metric. For a deeper algebraic and geometric perspective on the when and how of these structures, see G2-structure.
Topology, moduli, and invariants
Seven-dimensional manifolds with G2-structures connect geometry to topology in a direct way. The existence of a torsion-free G2-structure imposes strong topological constraints and yields a Ricci-flat metric, placing the manifold in a special position within the broader landscape of Riemannian geometry.
- Betti numbers b^k(M) play a key role in deformations. In particular, the space of deformations of a torsion-free G2-structure is controlled by harmonic 3-forms, giving a moduli space with dimension equal to b^3(M). This moduli space is typically smooth, reflecting the elliptic nature of the underlying deformation problem.
- The holonomy being contained in G2 places rigidity on the manifold’s curvature and affects possible geometric transitions, such as those encountered in gluing constructions or resolutions of singularities.
- In the context of physics, compactifications of higher-dimensional theories on G2-manifolds lead to lower-dimensional effective theories with specific supersymmetry properties, linking differential geometry to ideas from M-theory and related frameworks.
The topology of a compact G2-manifold is often studied through its cohomology, spin structure, and the interaction between φ and the manifold’s differential forms. The interplay between topology and geometric structures on M continues to be an active area of research, with ongoing questions about the full landscape of compact G2-manifolds and the connected components of their moduli spaces.
History and impact
- Berger’s foundational classification of possible holonomy groups set the stage for recognizing exceptional holonomy as a rich source of geometric structures. See holonomy and Berger classification.
- Bryant and Salamon’s explicit constructions in the late 1980s provided the first large family of complete non-compact G2-manifolds, demonstrating the viability of G2-geometry beyond formal existence results. See Bryant–Salamon.
- Dominic Joyce’s work in the mid-1990s and beyond produced the first known compact examples of manifolds with holonomy exactly G2, revolutionizing the field and inspiring numerous subsequent techniques. See Joyce, Dominic.
- The twisted connected sum method, introduced by Kovalev and developed by subsequent researchers, created a practical, scalable pathway to many new G2-manifolds and has influenced both the mathematics and the mathematical physics communities. See twisted connected sum.
The study of 7-manifolds and their G2-structures continues to intersect with algebraic geometry, differential topology, global analysis, and high-energy physics, making it a vibrant and evolving area of modern mathematics.