Absolutely split real algebraic vector bundles over a real form of projective space

Let M be a projective variety, defined over the field of real numbers, with the property that the base change MR_×C is isomorphic to CPN for some N. A real algebraic vector bundle E over M will be called absolutely split if the vector bundle ER_⊗C over MR_×C splits into a direct sum of line bundles. We classify the isomorphism classes of absolutely split vector bundles over M.     

Positive polynomials on projective limits of real algebraic varieties

We reveal some important geometric aspects related to non-convex optimization of sparse polynomials. The main result, a Positivstellensatz on the fibre product of real algebraic affine varieties, is iterated to a comprehensive class of projective limits of such varieties. This framework includes as necessary ingredients recent works on the multivariate moment problem, disintegration and projective limits of probability measures and basic techniques of the theory of locally convex vector spaces. A variety of applications illustrate the versatility of this novel geometric approach to polynomial optimization.     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

Automorphism groups of real algebraic curves which are double covers of the real projective plane

For each integer g≥ 3 we give the complete list of groups acting as full automorphism groups of real algebraic curves of genus $g$ which are double covers of the real projective plane. Explicit polynomial equations of such curves and the formulae of their automorphisms are also given. Received: 29 April 1999 / Revised version: 26 November 1999     

A lower bound for higher topological complexity of real projective space

We obtain an explicit formula for the best lower bound for the higher topological complexity, ${\mathrm{TC}}_k(R{P}^n)$, of real projective space implied by mod 2 cohomology.     

Stable constant mean curvature hypersurfaces in the real projective space

In this paper, we prove that the only compact two-sided hypersurfaces with constant mean curvature H which are weakly stable in and have constant scalar curvature are (i) the twofold covering of a totally geodesic projective space; (ii) the geodesic spheres in ; and (iii) the quotient to of the hypersurface obtained as the product of two spheres of dimensions k and n − k, with k = 1,..., n − 1, and radii r and , respectively, with .     

Totally real minimal 2-spheres in quaternionic projective space

Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(？) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(？)RPn(？) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (？) RPn(？) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.     

Compact minimal hypersurfaces with index one in the real projective space

Let M ⁿ be a compact (two-sided) minimal hypersurface in a Riemannian manifold . It is a simple fact that if has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.?We prove that if is the real projective space , obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface obtained as the product of two spheres of dimensions n ₁, n ₂ and radius R ₁, R ₂, with and . Received: June 6, 1998     

The moduli space of stable vector bundles over a real algebraic curve

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension of \mathbbZ/2{\mathbb{Z}/2} by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.     

The structure of projective maps between real projective manifolds

In this paper we study the set of projective maps between compact properly convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When domain is irreducible and the target is strictly convex, our results imply that each non-trivial homotopy class contains at most one projective map. These results are motivated by the theory of holomorphic maps between compact complex manifolds.