Birkhoff implies Quasi-topological

Quasi-topological gravities (QTGs) are higher-curvature extensions of Einstein gravity in $D\geq 5$ spacetime dimensions. Throughout the years, different notions of QTGs constructed from analytic functions of polynomial curvature invariants have been introduced in the literature. In this paper, we show that all such definitions may be reduced to three distinct inequivalent notions: type I QTGs, for which the field equations evaluated on a single-function static and spherically symmetric ansatz are second order; type II QTGs, whose field equations on general static and spherically symmetric backgrounds are second order; and type III QTGs, for which the trace of the field equations on a general background is second order. We show that type II QTGs are a subset of type I QTGs and that type III QTGs are a subset of type II QTGs modulo pure Weyl invariants. Moreover, we prove that type II QTGs possess second-order equations on general spherical backgrounds. This allows us to prove that any theory satisfying a Birkhoff theorem is a type II QTG, and that the reverse implication also holds up to a zero-measure set of theories. For every theory satisfying Birkhoff's theorem, the most general spherically symmetric solution is a generalization of the Schwarzschild spacetime characterized by a single function which satisfies an algebraic equation.