Frobenius manifold
In the mathematical field of differential geometry, a Frobenius manifold is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space. The concept generalizes the notion of Frobenius algebra to tangent bundles. They were introduced by Dubrovin.
Frobenius manifolds occur naturally in the subject of symplectic topology, more specifically quantum cohomology. The broadest definition is in the category of Riemannian supermanifolds. We will limit the discussion here to smooth (real) manifolds. A restriction to complex manifolds is also possible.
Let M be a smooth manifold. An affine flat structure on M is a sheaf Tf of vector spaces that pointwisely span TM the tangent bundle and the tangent bracket of pairs of its sections vanishes.
As a local example consider the coordinate vectorfields over a chart of M. A manifold admits an affine flat structure if one can glue together such vectorfields for a covering family of charts.
Let further be given a Riemannian metric g on M. It is compatible to the flat structure if g(X, Y) is locally constant for all flat vector fields X and Y.
A Riemannian manifold admits a compatible affine flat structure if and only if its curvature tensor vanishes everywhere.
A family of commutative products * on TM is equivalent to a section A of S2(T*M) ⊗ TM via
We require in addition the property
Therefore the composition g#∘A is a symmetric 3-tensor.
This implies in particular that a linear Frobenius manifold (M, g, *) with constant product is a Frobenius algebra M.
Given (g, Tf, A), a local potential Φ is a local smooth function such that
for all flat vector fields X, Y, and Z.
A Frobenius manifold (M, g, *) is now a flat Riemannian manifold (M, g) with symmetric 3-tensor A that admits everywhere a local potential and is associative.
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