Geometry of the Batalin–Fradkin–Vilkovisky theorem

Geometry of the BFV Theorem

We describe gauge-fixing at the level of virtual paths in the path integral as a nonsymplectic BRST-type of flow on the path phase space. As a consequence a gauge-fixed, non-local symplectic structure arises. Restoring of locality is discussed. A pertinent anti-Lie-bracket and an infinite dimensional group of gauge fermions are introduced. Generalizations to Sp(2)-symmetric BLT-theories are mad...

The fundamental theorem of projective geometry

We prove the fundamental theorem of projective geometry. In addition to the usual statement, we also prove a variant in the presence of a symplectic form.

The Fundamental Theorem of q-Clan Geometry

CONFORMAL GEOMETRY SEMINAR The Poincaré Uniformization Theorem

1.1. Geometry. A covariant derivative on a manifold M is an operator ∇XY on vector fields X and Y satisfying for any smooth function f : (i) ∇fXY = f∇XY ; and (ii) ∇X(fY ) = f∇XY + (∇Xf)Y . If g is a Riemannian metric on M , then there is associated with g a unique covariant derivative ∇ characterized by: (iii) ∇XY −∇YX = [X,Y ]; and (iv) ∇X ( g(Y,Z) ) = g(∇XY, Z) + g(Y,∇XZ). We define the Chri...

The Three Gap Theorem and Riemannian Geometry

The classical Three Gap Theorem asserts that for n ∈ N and p ∈ R, there are at most three distinct distances between consecutive elements in the subset of [0, 1) consisting of the reductions modulo 1 of the first n multiples of p (see e.g. [Só58], [Św58]). There are several interesting generalizations of this theorem in [FrSó92], [Vi08], and the references therein. Regarding it as a statement a...