Differential Geometry

For a given real generic curve $\ga: S^1\to \Bbb {RP}^n$ let $D_\ga$ denote the ruled hypersurface in RP n consisting of all osculating subspaces to $\ga$ of codimension 2. A curve $\ga: S^1\to \Bbb {RP}^n$ is called convex if the total number of its intersection points (counted with multiplicities) with any hyperplane in RP n does not exceed n . In this short note we show that for any two convex real projective curves $\ga_1:S^1\to\Bbb {RP}^n$ and $\ga_2:S^1\to\Bbb {RP}^n$ the pairs $(\Bbb {RP}^n,D_{\ga_1})$ and $(\Bbb {RP}^n,D_{\ga_2})$ are homeomorphic answering a question posed by V.Arnold.

The wall-crossing formula for Donaldson invariants of smooth, simply connected four manifolds with b + =1 is shown to be a topological invariant of the manifold for reducible connections with two or fewer singular points. The explicit formulas derived agree with those of Ellingsrud and Gottische and Friedman and Qin for algebraic manifolds.

We relate the Donaldson invariants of two four-manifolds X i with embedded Riemann surfaces of genus 2 and self-intersection zero with the invariants of the manifold X which appears as a connected sum along the surfaces. When the original manifolds are of simple type with b 1 =0 and b + >1 , X is of simple type with b 1 =0 and b + >1 as well, and the relationship between the invariants is expressed as constraints in the basic classes for X. Also we give some applications. For instance, if X i have both b 1 =0 then X is of simple type with b 1 =0 , b + >1 , and has no basic classes evaluating zero on the Riemann surface. Finally, we prove that any four-manifold with b + >1 and with an embedded surface of genus 2, self-intersection zero and representing an odd homology class, is of finite type of second order.

We prove that every suitable 4 -manifold with b 1 =0 and with an embedded Riemann surface of genus 2 is of simple type. We find a relationship between the basic classes of two of these 4 -manifolds and those of the connected sum along the Riemann surface.

We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either LA--groupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We also show that the cotangent of a double Lie groupoid is a symplectic double groupoid, and that the side groupoids of any symplectic double groupoid are Poisson groupoids in duality. Thus any double Lie groupoid gives rise to a dual pair of Poisson groupoids.

The most important examples of a double vector bundle are provided by iterated tangent and cotangent functors: TTM, TT^*M, T^*TM, and T^*T^*M. We introduce the notions of the dual double vector bundle and the dual double vector bundle morphism. Theorems on canonical isomorphisms are formulated and proved. Several examples are given.

We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support respectively. We show explicitly that the latter spaces are asymptotically isomorphic to the former. We discuss briefly quantisation of Teichm\" uller spaces and some other application of the constructed approach. The paper does not require any preliminary knowledge of the subject above the Poincar\' e uniformisation theorem.

We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincaré duality in Lie algebra cohomology and de Rham cohomology. We show that the duality isomorphisms can be given by a cap product with a suitable fundamental class and hence may be taken natural in any reasonable sense. Thereafter, for a Lie-Rinehart algebra satisfying duality, we introduce a certain intrinsic module which provides a crucial ingredient for the construction of the bilinear pairings. This module determines a certain class, called modular class of the Lie-Rinehart algebra, which lies in a certain Picard group generalizing the abelian group of flat line bundles on a smooth manifold. Finally, we show that a Poisson algebra having suitable properties determines a certain module for the corrresponding Lie-Rinehart algebra and hence modular class whose square yields the module and characteristic class for its Lie-Rinehart algebra mentioned before. In particular, this gives rise to certain bilinear pairings in Poisson homology.

We study the link between a compact hypersurface in ¶ n+1 and the set of all its tangent planes. In this context, we identify ¶ n+1 to the set of linear subspaces of codimension one by orthogonal complementarity. This gives rise to a kind of duality which has already been studied Bruce and Romerro-Fuster, and relates a hypersurface to the set of its tangent planes. But in these papers the dual, in this sense, of the set of tangent planes of a hypersurface was not defined and iteration of the procedure was not possible. Therefore we extend this type of duality to more general sets and achieve a procedure which can be iterated and gives in fact an involution.

We consider the Dirac operator on compact quaternionic Kaehler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.