Relational particle dynamics include the dynamics of pure shape and cases in which absolute scale or absolute rotation are additionally meaningful. These are interesting as regards the absolute versus relative motion debate as well as discussion of conceptual issues connected with the problem of time in quantum gravity. In spatial dimension 1 and 2 the relative configuration spaces of shapes ar...

Introduction A map : M ! N of Riemannian manifolds is harmonic if it extremises the energy functional: Z jdj 2 dvol on every compact subdomain of M. Harmonic maps arise in many diierent contexts in Geometry and Physics (for an overview, see 15,16]) but the setting of concern to us is the following: take M to be 2-dimensional and N to be a Riemannian symmetric space of compact type. In this case...

Among all complex projective varieties X →֒ P(V ), the equivarient embeddings of homogeneous varieties—those admitting a transitive action of a semi-simple complex algebraic group G—are the easiest to study. These include projective spaces, Grassmannians, non-singular quadrics, Segre varieties, and Veronese varieties. In Joe Harris’ book “Algebraic Geometry: A First Course” [H], he computes the ...

It is known that any two points in Teichmüller space are joined by an earthquake path. In this paper we show any earthquake path R → T (S) extends to a proper holomorphic mapping of a simplyconnected domain D into Teichmüller space, where R ⊂ D ⊂ C. These complex earthquakes relate Weil-Petersson geometry, projective structures, pleated surfaces and quasifuchsian groups. Using complex earthquak...

1. Arnold's conjecture. An automorphism ^ of a symplectic manifold (P,u;) is homologous to the identity if there is a smooth family tyt (* € [0,1]) of automorphisms such that the time-dependent vector field £t defined by di^t/dt = & ° ^t is globally hamiltonian; i.e. if there is a smooth family Ht of real-valued functions on P such that £*JCÜ = dHt. It was conjectured by Arnold [1], as an exten...

We will prove that if an open subset of CPn is isometrically immersed into CPm, withm < (4/3)n−2/3, then the image is totally geodesic. We will also prove that if an open subset of HPn isometrically immersed into HPm, with m < (4/3)n− 5/6, then the image is totally geodesic.

We derive an explicit expression for an associative ∗-product on the fuzzy complex projective space, CPN−1 F . This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on CPN−1 F , represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU(N). In the li...

The standard model fermion spectrum, including a right handed neutrino, can be obtained as a zero-mode of the Dirac operator on a space which is the product of complex projective spaces of complex dimension two and three. The construction requires the introduction of topologically non-trivial background gauge fields. By borrowing from ideas in Connes’ non-commutative geometry and making the com...

The shape-space l.m whose points a represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric. When m = 1, we find that Y\ = S~; when m ^ 3, the shape-space contains singularities. This paper deals mainly with the case m = 2, when the shape-space I* c a n be identified with a version of CP*~. Of special importance are the shape-m...

Let V be an infinite-dimensional complex Banach space and X ⊂ P(V ) a closed analytic subset with finite codimension. We give a condition on X which implies that X is a complete intersection. We conjecture that the result should be true for more general topological vector spaces.