manuscripta mathematica

 Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B ? A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C 2 -module for this action, then B is a Frobenius algebra. For a fixed commutative Frobenius algebra B of finite dimension n equipped with an involution σ, we characterize the central simple algebras A of degree n that contain B and carry involutions extending σ.

Let us consider M a closed smooth connected m-manifold, N a smooth (2m − 2)-manifold and $${ f : M \longrightarrow N}$$ a continuous map, with $${ m \equiv 1(4)}$$ . We prove that if $${ {f}_* : {H}_{1}(M; \, Z_2) \longrightarrow \check{H}_{1}(f(M) ; \, Z_2)}$$ is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104, 97–110, 2001; Koschorke in The singularity method and immersions of m-manifolds into manifolds of dimensions 2m − 2, 2m − 3 and 2m − 4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281–285, 1992).

On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one.

We derive in a simple way some higher estimates of the number of every-where linearly independent cross-sections for even multiples of real vector bundles, and, more generally, for a certain class of homomorphism bundles, supposing that there exists at least one nowhere vanishing cross-section.

In this paper we study the algebra structure of the cohomology ring of a monomial algebra.

It is shown that 5906 is the. least integer expressible as the sum of two rational fourth powers but not as the sum of two integer fourth powers. The relevant Diophantine equation x4+y4=D represents a curve of genus 3, and extensive arithmetic calculations are involved: in particular, class-number, units and ideal-class stucture are. computed for four particular eighth degree extension fields of the rationals. The result provides several examples of curves of genus 3, everywhere locally solvable, but with no rational points.

We extend the definition of an m-stable curve introduced by Smyth to the setting of maps to a projective variety X, generalizing the definition of a Kontsevich stable map in genus one. We prove that the moduli problem of n-pointed m-stable genus one maps of class β is representable by a proper Deligne–Mumford stack $${\overline{\mathcal {M}}_{1,n}^{m}(X,\beta)}$$ over Spec $${\mathbb {Z}[1/6]}$$ . For $${X=\mathbb {P}^{r},}$$ we show that $${\overline{\mathcal {M}}_{1,n}^{m}(\mathbb {P}^{r},d)}$$ is irreducible for m sufficiently large. We also show that $${\overline{\mathcal {M}}_{1,n}^{m}(\mathbb {P}^r,d)}$$ is smooth if d + n ≤ m ≤ 5.