Ricci flow and the holonomy group

TL;DR

This paper proves that the restricted holonomy group of a complete Ricci flow solution with bounded curvature remains unchanged over time, ensuring geometric properties like Kähler or reducible structures are preserved at singularities.

Contribution

It establishes that the holonomy group cannot spontaneously contract under Ricci flow, confirming its invariance and linking geometric structures at finite singular times.

Findings

Holonomy group remains invariant under Ricci flow with bounded curvature.

Kähler and reducible structures are preserved at finite singular times.

Holonomy contraction cannot occur spontaneously in finite time.

Abstract

We prove that the restricted holonomy group of a complete smooth solution to the Ricci flow of uniformly bounded curvature cannot spontaneously contract in finite time; it follows, then, from an earlier result of Hamilton that the holonomy group is exactly preserved by the equation. In particular, a solution to the Ricci flow may be K\"{a}hler or locally reducible (as a product) at $t= T$ if and only if the same is true of $g(t)$ at times $t\leq T$.