Approximation on Nash sets with monomial singularities

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash sets  X⊂M$X\subset M$whose singularities are monomial. To that end we discuss first finiteness and weak normality for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space, and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k   semialgebraically diffeomorphic, for k>m²$k>m^2$, are also Nash diffeomorphic.     

Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p∈M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff convergence theory, that, if model volume growth is sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional space. Hence, our main theorem has various advantages of the Cheeger-Colding diffeomorphism theorem via the Euclidean volume growth. Our main theorem also contains a result of do Carmo and Changyu as a special case.     

Detecting codimension one manifold factors with topographical techniques

We prove recognition theorems for codimension one manifold factors of dimension n?4. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that X×R is a manifold in the cases when X is a resolvable generalized manifold of finite dimension n?3 with either: (1) the crinkled ribbons property; (2) the twisted crinkled ribbons property and the disjoint point disk property; or (3) the fuzzy ribbons property.     

Embedding solenoids in foliations

In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H¹(L;R) not equal to 0 and if the foliation is a product foliation in some saturated open neighborhood U of L, then there exists a foliation F^′ on M which is C¹-close to F, and F^′ has an uncountable set of solenoidal minimal sets contained in U that are pairwise non-homeomorphic. If H¹(L;R) is 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighborhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg.     

Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x₁,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T_•(Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ? over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T_•(?) that is a resolution of Ker?. We also show that the length of T_•(?) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ?.     

Conformal deformations of integral pinched 3-manifolds

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S³. The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.     

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.     

On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

It is well known that if the tangent bundle TM of a Riemannian manifold (M,g) is endowed with the Sasaki metric gs, then the flatness property on TM is inherited by the base manifold [Kowalski, J. Reine Angew. Math. 250 (1971) 124-129]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace gs by the most general Riemannian “g-natural metric” on TM (see [Kowalski and Sekizawa, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1-29; Abbassi and Sarih, Arch. Math. (Brno), submitted for publication]). In this direction, we prove that if (TM,G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M,g) possesses the same property, respectively. We also give explicit examples of g-natural metrics of arbitrary constant scalar curvature on TM.     

Extrinsic radius pinching for hypersurfaces of space forms

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L^∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L^∞-norm.     

On three-dimensional conformally flat quasi-Sasakian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O₁]     

Lorentzian similarity manifolds

An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝ ^m+2), where G = ℝ ^m+2 ⋊ (O(m+1, 1)×ℝ⁺). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S ^m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.     

On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengths of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M,θ). Any Sasakian manifold (M,θ) whose pseudohermitian sectional curvature Kθ(σ) is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of φ-sectional curvature c=−3. We show that the Lie algebra i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n has dimension ?²(n+1) and if dimRi(M,θ)=²(n+1) then Hθ(σ)= constant.     

On an existence theorem of Wagner manifolds

In their common paper [An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 43 (1997) 307-321] the authors give a condition for a 1-form β to be the perturbation of a Riemannian manifold (M, α) such that the manifold equipped with any (α, β) -metric is a Wagner manifold with respect to the Wagner connection induced by β. The condition shows that its covariant derivative with respect to the Lévi-Civita connection must be of a special form

(∇β)(X,Y)=∥β^#∥2α(X,Y)−β(X)β(Y)     

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Let ${\mathcal{D}}$ be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the ?? invariant ???= ??(M) for any orientable compact flat manifold M. After giving an explicit expression for ??(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining ??(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p??? 7, ??(s) can be expressed as a multiple of L _??(s), an L-function associated to a quadratic character mod p, while ??(0) is a (non-zero) integral multiple of the class number h _?p of the number field ${\mathbb Q(\sqrt {-p})}$ . In the case of metacyclic groups of odd order pq, with p, q primes, we show that ??(0) is a rational multiple of h _?p .     

Nonpositive curvature: A geometrical approach to Hilbert-Schmidt operators

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that GM, the Banach-Lie group generated by M, acts isometrically and transitively on M. Moreover, GM admits a polar decomposition relative to M, namely GM?M×K as Hilbert manifolds (here K is the isotropy of p=1 for the action ), and also GM/K?M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism NM?Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed ea, we obtain ea=exevex with ex∈M and v orthogonal to M at p=1. As a corollary we obtain decompositions for the full group of invertible elements G?M×exp(T₁M^⊥)×K.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this paper, we provide a lower bound for h⁰(m(K_X+L)) under the assumption that κ(K_X+L)≥0. In particular, we get the following: (1) if 0≤κ(K_X+L)≤2, then h⁰(K_X+L)>0 holds. (2) If κ(K_X+L)=3, then h⁰(2(K_X+L))≥3 holds. Moreover we get a classification of (X,L) with κ(K_X+L)=3 and h⁰(2(K_X+L))=3 or 4.     

A new variational characterization of the Wulff shape

Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.     

On Fox?s m-dimensional category and theorems of Bochner type

We show that catm(X)=cat(jm), where catm(X) is Fox?s m-dimensional category, jm:X→X[m] is the mth Postnikov section of X and cat(X) is the Lusternik-Schnirelmann category of X. This characterization is used to give new “Bochner-type” bounds on the rank of the Gottlieb group and the first Betti number for manifolds of non-negative Ricci curvature. Finally, we apply these methods to obtain Bochner-type theorems for manifolds of almost non-negative sectional curvature.