The space of inequivalent representations of a compact surface S with χ(S) < 0 as a quotient of a convex domain in RP by a properly discontinuous group of projective transformations is a cell of dimension

/pdf/convex-real-projective-structures-on-compact-surfaces-1p24ztrtk0.pdf

Convex real projective structures on compact surfaces

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

https://arxiv.org/pdf/math-ph/0509016.pdf

/pdf/projective-structures-with-fuchsian-holonomy-3uwsv2eh63.pdf

Projective structures with Fuchsian holonomy

An affine manifold is a differentiable manifold together with an atlas of coordinate charts whose coordinate changes extend to affine automorphisms of Euclidean space. These charts are called atline coordinates. A map between affine manifolds is called affine it its expression in affine coordinates is the restriction of an aftine map between vector spaces. Thus we form the category of affine manifolds and affine m_nps. Let M be a connected affine manifold of dimension n-> 1, locally isomorphic to the vector space E. Its universal covering/~/ inheri ts a unique affine structure for which the covering projection/~/--~ M is an aifine immersion. The group ~r of deck transformations acts on /V/by afline automorphisms. It is well known that there is an affine immersion D :/~5/--~ E, called the developing map. This follows, for example, from Chevalley's Monodromy Theorem; a proof is outlined in Section 2. Such an immersion is unique up to composition with an atiine automorphism of E. Thus for every g ~ "n" there is a unique affine automorphism a(g) of E such that D o g =ct(g)oD. The resulting homomorphism a : ~r --~ Aft (E) from 7r into the group of affine automorphisms of E is called the affine holonomy representation. It is unique up to inner automorphisms of Aft (E). The composition A : 7r ~ G L (E) is called the linear holonomy representation. The affine structure on M is completely determined by the pair (D, a) . M is called complete when D is a homeomorphism. This is equivalent to geodesic completeness of the connection on M (in which parallel transport is locally defined by affine charts as ordinary parallel transport in E). It is notorious that compactness does not imply completeness. The main results of this paper are about aftine manifolds whose affine holonomy groups a(Tr) are nilpotent. An important class of such manifolds are the affine nilmanifolds 7r\G. Here 7r is a discrete subgroup of a simply connected nilpoint Lie group G. It is assumed that G has a left-invariant afline structure; the space of right cosets of r then inherits an affine structure.

/pdf/affine-manifolds-with-nilpotent-holonomy-21ydyjh3d9.pdf

Affine manifolds with nilpotent holonomy

In these two papers we intend to study the space of all affine structures on the real 2-dimensional torus T, a problem suggested by C. Ehresmann in 1936, or more specifically by S. S. Chern in one of his lectures and attacked by N. H. Kuiper [6] among others. An affine structure on a mainfold is a maximal atlas whose coordinate transformations belong to the affine transformation group A(ri) on the affine space. Our main purpose is to describe the set {Γ} of all affine structures on T module the group Diίf \\T2]e\\ here Diff [T ],, is the group of all diίfeomorphisms of T which induce the identity on the fundamental group π^T). The space {r}/Diff[T2]e, equipped with an appropriate topology, is regarded as an affine version of the Teichmϋller space. In the usual case the holonomy group H of an affine structure on a mainfold is defined as a subgroup of the affine transformation group A(n) up to the conjugate class. In this work, however, we construct a modified holomony group if* for an affine structure so that in the case of 2-dimensional affine torus the group H* is a subgroup ofA(2)e, the universal covering group of the identity component of A(2). We do this in such a way that the modified holonomy group H* is mapped onto the usual holonomy group by the projection mapping. With this modification of the holonomy group the first main result in the paper could be summarized as follows (Theorem 3.3 and 4.15): the affine structures on T are completely determined by their modified holonomy groups #*. Carrying out the determination of holonomy groups Jΐ*, we describe the space {Γh} IΌiff[T 2]e of the homogeneous affine structures on T . As Y. Matsushima [7] discusses for complex tori in a somewhat different way, we show the following (Theorem 3.10): the space {ΓA}/DiίF[Γ 2]β is an affine algebraic or, more precisely a 4-dimensional quadratic cone in R without singularities variety, except at the vertex, the vertex itself corresponding to the natural affine

/pdf/the-affine-structures-on-the-real-two-torus-i-3z9rrve165.pdf

The affine structures on the real two-torus. I

/pdf/on-compact-homogeneous-affine-manifolds-3dv35i9wh5.pdf

On compact homogeneous affine manifolds

/pdf/super-differential-calculus-9tkwi1gnfc.pdf

Super differential calculus

/pdf/the-affine-structures-on-the-real-two-torus-i-330hwt9mih.pdf

Katsumi Yagi

Papers

The affine structures on the real two-torus. I

On compact homogeneous affine manifolds

Super differential calculus

The affine structures on the real two-torus. I