Formal manifold explained

In geometry and topology, a formal manifold can mean one of a number of related concepts:

• In the sense of Dennis Sullivan, a formal manifold is one whose real homotopy type is a formal consequence of its real cohomology ring; algebro-topologically this means in particular that all Massey products vanish.^[1]
• A stronger notion is a geometrically formal manifold, a manifold on which all wedge products of harmonic forms are harmonic.^[2]

References

1. Book: Sullivan, Dennis. Dennis Sullivan

   . Dennis Sullivan. Differential forms and the topology of manifolds. Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) . 37–49. University of Tokyo Press. Tokyo. 1975. 0370611. 0319.58005.

2. Kotschick. Dieter. Dieter Kotschick. On products of harmonic forms. Duke Mathematical Journal. 107. 3. 521–531. 2001. 1828300. 10.1215/S0012-7094-01-10734-5. math/0004009.