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On products of harmonic forms. (English) Zbl 1036.53030

A closed manifold is said to be geometrically formal if it admits a Riemannian metric for which all wedge products of harmonic forms are harmonic; such a metric is called a formal Riemannian metric. Examples are compact globally symmetric spaces and rational homology spheres. In D. Sullivan [Manifolds, Proc. Int. Conf., Tokyo, 1973 (1975; ; Zbl 0319.58005)] it was seen that in many cases the rational homotopy type is a formal consequence of the cohomology ring. The notion of geometric formality is stronger than Sullivan’s notion of formality in this sense.
The paper gives a number of elementary topological obstructions to geometric formality of closed oriented manifolds. These obstructions for dimension \(\leq4\), are enough to force geometrically formal manifolds to have the same real cohomology algebra as (though not necessarily be homotopic to) a compact globally symmetric space. In dimension 2, non-vanishing of \(b_1(\Sigma).\chi(\Sigma)\) obstructs geometric formality. On a 2-torus, formal Riemannian metrics must be flat.
Suppose \(M\) is a closed oriented \(n\)-dimensional geometrically formal manifold. The paper gives a number of limitation for the Betti numbers. For example, \(b_K(M)\leq{}b_k({\mathbf T}^n)\); \(b_1(M)\not=n-1\), while if \(n=4m\) then \(b^\pm_{2m}(M)\leq b^\pm_{2m}({\mathbf T}^n)\). Furthermore, if \(b_1(M)=k\), then there is a smooth submersion \(\pi:{}M\rightarrow{\mathbf T}^k\) inducing an injection of the cohomology algebras. When \(b_1(M)=n\), \(M\) is forced to be diffeomorphic to \({\mathbf T}^n\) while all formal Riemannian metrics are flat. Conversely, a closed oriented \(n\)-dimensional manifold which fibers smoothly over \(S^1\) with \(b_1=1\) and \(b_k=0\) for all \(1<k<n-1\) is guaranteed to be geometrically formal.
In dimension 3 a manifold with \(b_1=1\) is geometrically formal precisely when it fibers over \(S^1\). In dimension 4, the paper discusses constraints on \(b_1\) and the cohomology ring, looking at each of the cases \(b_1=0,1,2,4\) in turn. For example, a closed oriented 4-manifold with \(b_1=1\) and \(b_2=0\) is geometrically formal iff it fibers over \(S^1\); from here an example of a non-symmetric geometrically formal manifold appears as an \(S^1\times{}M\) where \(M\) is a non-symmetric rational homology sphere.
The paper goes on to give some obstructions from symplectic geometry. The simplest of these requires a closed oriented geometrically formal 4-manifold with \(b_2^+>0\) to admit a symplectic structure; from here an example can be constructed which is not geometrically formal but for which this cannot be detected from its real cohomology ring.
The paper ends with a result showing that even the few geometrically formal manifolds usually admit nonformal metrics. In fact a closed oriented manifold admits a nonformal metric iff it is not a rational homology sphere.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
57R57 Applications of global analysis to structures on manifolds
58A14 Hodge theory in global analysis

Citations:

Zbl 0319.58005
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References:

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