[en] The question considered is whether or not a Riemannian metric can be found to make a given curve field on a closed surface into geodesics. Allowing singularities removes the restriction to Euler characteristic zero. The main results are the following: only two types of isolated singularities can occur in a geodesic field on a surface. No geodsic fields exist on a surface with Euler characteristic less than zero. If the Euler characteristic is zero, such a geodesic field can have only removable singularities. Only a limited number of geodesic fields exist on S² and RP². A closed geodesic (perhaps made from several curves and singularities) always appears in such a field

[en] This note follows a previous note on the existence of space-like maximal submanifolds in a hyperbolic riemannian differentiable manifold. It contains some uniqueness and non-existence theorems, and a study of the maximisation of area by such a submanifold

[en] For conservative linear homogeneous nonholonomic systems, there exists a cotangent bundle with the symplectic structure dπ^μ ∧ dξ_μ, in which the motion equations of the system can be written into the form of the canonical equations by the set of quasi-coordinates π^μ and quasi-momenta ξ_μ. The key to construct this cotangent bundle is to define a set of suitable quasi-coordinates π^μ by a first-order linear mapping, so that the reduced configuration space of the system is a Riemann space with no torsion. The Hamilton–Jacobi method for linear homogeneous nonholonomic systems is studied as an application of the quasi-canonicalization. The Hamilton–Jacobi method can be applied not only to Chaplygin nonholonomic systems, but also to non-Chaplygin nonholonomic systems. Two examples are given to illustrate the effectiveness of the quasi-canonicalization and the Hamilton–Jacobi method. (paper)

[en] The hypersurfaces of Eⁿ⁺¹ have been studied for the particular case when they satisfy the R.C-condition or the C.R-condition. One objective is to generalize this situation to a higher codimension. More precisely, we consider the case of dimension 4, and replace the condition of quasiumbilicity by the conformal flatness. In this way, we construct an example of 4-submanifold of IE⁶ which is conformally flat at a particular point without being quasiumbilical. That such submanifolds exist, was asserted without proof. Thus, we present another counter-example. (author). 4 refs

[en] The boundary behaviour of classes of ring mappings, which generalize quasiconformal mappings in the sense of Gehring, is under investigation. Theorems proving that they have continuous boundary extensions are established in terms of prime ends of regular domains. Results on the equicontinuity of mappings in these classes in the closure of a fixed domain are also established in these terms.

Bibliography: 45 titles. (paper)

[en] Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.

[en] Every branched superminimal surface of area 4πd in S⁴ is shown to arise from a pair of meromorphic functions (f₁,f₂) of bidegree (d,d) such that f₁ and f₂ have the same ramification divisor. Conditions under which branched superminimal surfaces can be generated from such pairs of functions are derived. For each d ≥ 1 the space of harmonic maps (i.e branched superminimal immersions) of S² into S⁴ of harmonic degree d is shown to be a connected space of complex dimension 2d+4. (author). 18 refs

[en] 1. Cohomology of vector bundles and the duality theorem. 2. Divisors, line bundles and the Riemann-Roch theorem. 3. Projective embedding of a compact Riemann surface. 4. Genus and first Betti number. 5. Chern class and degree. 6. The Jacobian. 7. Line bundles and characters. 8. Poincare bundle. 9. The Picard manifold of a compact Kaehler manifold. 10. Vector bundles on a compact Riemann surface. 11. The Riemann-Roch theorem for vector bundles. 12. Indecomposable bundles and the Krull-Remak-Schmidt theorem. 13. Weil's theorem; unitary bundles. Appendix: Factors of automorphy. (author)

[en] A foliation that admits a Weyl structure arising from a pseudo-Riemannian metric of any signature as its transverse structure is called a pseudo-Riemannian Weyl foliation or (for short) a Weyl foliation. We investigate codimension q ≥ 2 Weyl foliations on (not necessarily compact) manifolds. Different interpretations of their holonomy groups are given. We prove a criterion for a Weyl foliation to be pseudo-Riemannian. We find a condition on the holonomy groups which guarantees the existence of a transitive attractor of (M, F ). Moreover, if the Weyl foliation is complete, this condition implies the existence of a global transitive attractor. We describe the structure of complete Weyl foliations modelled on Riemannian manifolds. (paper)

[en] In this paper, we will prove vanishing and finiteness theorems for $L^2$-harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. From these theorems and the work of Li–Tam, we can obtain some one-end and finite ends results for the locally conformally flat Riemannian manifold.