Graded metrics adapted to splittings

Homogeneous graded metrics over split ℤ₂-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry: It amounts to determining a graded metric by the data {g, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with κ ∈ Hom (TM, E) invertible, and Δ′κ = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only over a Riemannian manifold of constant curvature, and the curvature of Δ′ onE satisfiesR ^Δ′ (X,Y)² = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection Δ′ satisfies a specific set of equations. 0 Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Ω(M) are also discussed. Partially supported by DGICYT grants #PB94-0972, and SAB94-0311; IVEI grant 95-031; CONACyT grant #3189-E9307.     

A characterization of quintic helices

A polynomial curve of degree 5, α,$\alpha ,$ is a helix if and only if both ∥α^′∥$\parallel {\alpha }^{\prime }\parallel$ and ∥α^′∧α^″∥$\parallel {\alpha }^{\prime }\wedge {\alpha }^{″}\parallel$ are polynomial functions.     

A third order partial differential equation for isotropic boundary based triangular Bézier surface generation

We approach surface design by solving a linear third order Partial Differential Equation (PDE). We present an explicit polynomial solution method for triangular Bézier PDE surface generation characterized by a boundary configuration. The third order PDE comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator over triangular Bézier surfaces.     

A characterization of helical polynomial curves of any degree

We give a full characterization of helical polynomial curves of any degree and a simple way to construct them. Existing results about Hermite interpolation are revisited. A simple method to select the best quintic interpolant among all possible solutions is suggested.      

The distance trisector curve is transcendental

We show that the distance trisector curve is not an algebraic curve, as was conjectured in the founding paper by Asano et al. (Adv Math 212:338–360, 2007).     

The exterior derivative as a Killing vector field

Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms. Partially supported by DGICYT grants #PB91-0324, and SAB94-0311; CONACyT grant #3189-E9307.     

PDE triangular Bézier surfaces: Harmonic, biharmonic and isotropic surfaces

We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a fourth-order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular Bézier surfaces.     

Total stereochemical control in the addition of thiols to p-toluenesulfonylacetylene. synthesis of Z- and E-2-sulfanylvinylsulfonyl derivatives

Alkyl and aryl sulfides react with equimolecular amounts of p-toluenesulfonylacetylene in CH3CN at 0 degrees C or rt without the use of any catalytic reagent to give good yields of Z-2-sulfanylvinylsulfonyl derivatives with total diastereoselectivity. On the other hand, in the presence of 1.1 equiv of NaH in THF, the same reaction affords the corresponding E-diastereomer also with total diastereoselectivity.     

Construction of non-hyperspherical immersions

We give a method using the complex functions associated to the Gauss map to construct examples of non-hyperspherical immersions of ?² in R ⁴ with two globally defined orthogonal families of asymptotic lines having an isolated inflection point.