On minor-minimally 3-connected binary matroids

A matroid M is called minor-minimally 3-connected if M is 3-connected and, for each e∈E(M), either M?e or M/e is not 3-connected. In this paper, we prove a chain theorem for the class of minor-minimally 3-connected binary matroids. As a consequence, we obtain a chain theorem for the class of minor-minimally 3-connected graphs.     

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.     

Almost Schur lemma for manifolds with boundary

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.     

A new proof of the theorem: Harmonic manifolds with minimal horospheres are flat

In this note we reprove the known theorem: Harmonic manifolds with minimal horospheres are flat. It turns out that our proof is simpler and more direct than the original one. We also reprove the theorem: Ricci flat harmonic manifolds are flat, which is generally affirmed by appealing to Cheeger–Gromov splitting theorem. We also confirm that if a harmonic manifold M has same volume density function as ? ⁿ , then M is flat.     

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M ⁿ is simply connected and the k-th Ricci curvature of M ⁿ is bounded below by a quantity involving the mean curvature of M ⁿ and the curvature of the ambient manifold, then M ⁿ is diffeomorphic to the standard sphere ${\mathbb{S}^n}$ . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M ⁿ is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.     

Relative formality theorem and quantisation of coisotropic submanifolds

We prove a relative version of Kontsevich's formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich's theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L_∞-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevich's theorem for supermanifolds, which we also discuss. In physical language, the result states that there is a duality between the Poisson sigma model on a manifold with a D-brane and the Poisson sigma model on a supermanifold without branes (or, more properly, with a brane which extends over the whole supermanifold).     

Moufang loops of odd prime exponent p and a class of linear spaces of order p

In this paper it is shown that any finite Moufang loop M of odd prime exponent p gives rise to a linear space Σ(M) with a large number of affine desarguesian planes, and a theorem on the structure of Σ(M) is proved.     

Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M₊ and M₋, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.     

Finding a small 3-connected minor maintaining a fixed minor and a fixed element

LetN andM be 3-connected matroids, whereN is a minor ofM on at least 4 elements, and lete be an element ofM and not ofN. Then, there exists a 3-connected minor \(\bar M\) ofM that usese, hasN as a minor, and has at most 4 elements more thanN. This result generalizes a theorem of Truemper and can be used to prove Seymour’s 2-roundedness theorem, as well as a result of Oxley on triples in nonbinary matroids.     

Surgery and harmonic spinors

Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.     

The higher rank rigidity theorem for manifolds with no focal points

We say that a Riemannian manifold M has rank M ≥ k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns–Spatzier, later generalized by Eberlein–Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank k ≥ 2 (the “higher rank” assumption) whose isometry group Γ satisfies the condition that the Γ-recurrent vectors are dense in SM is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein–Heber to prove a generalization of this theorem where the manifold M is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann–Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.     

Uniqueness for algebraically regular solutions to pathological differential equations

A uniqueness theorem is proved for algebraically regular solutions to the unbounded initial value problem P′ = AP, P(0) = diag(1, 1, 1,…) in the real Banach algebra of infinite matrices $\text{M}$ with standard norm. It is not assumed that $\text{A}$ ∈ $\text{M}$, but it is required that A have an inverse in $\text{M}$, a property which is seen to be implied quite naturally by certain divergent or pathological systems. The conditions for the theorem are motivated by a particular system, previously considered by Hille and Feller, which arises from a divergent, purebirth, time dependent stochastic process, although no restriction requiring the solution matrix to be either stochastic or substochastic is necessary.The theorem may be easily generalized to any Banach algebra with identity.     

The Bakry–Emery Ricci tensor and its applications to some compactness theorems

Let (M, g) be a complete and connected Riemannian manifold of dimension n. By using the Bakry–Emery Ricci curvature tensor on M, we prove two theorems which correspond to the Myers compactness theorem.     

Another Rigidity Theorem for Affine Immersions

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: M ⁿ → Rⁿ⁺¹ be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of M ⁿ into R ^{n +} 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.     

On the Hahn--Mazurkiewicz theorem

The following generalization of the Hahn-Mazurkiewicz theorem is proved: Let (E,e) be a locally compact locally connected metric space. Let M be a continuum in this space and let d,e∈ M. Then there is a continuous mapping f: [0,1]→E such that f(0) = d, f(1)= e and M⊂f([0,1]). Also some corollaries of this theorem are proved.     

Hodge theorem for the natural projection of complex horizontal Laplacian on complex Finsler manifolds

Let M be a compact complex manifold with a complex Finsler metric F. We define a natural projection of complex horizontal Laplacian on M: it is independent of the fiber coordinate. By using Sobolev space theory and spectral resolution theory in Hilbert space, we prove the Hodge theorem for the natural projection of complex horizontal Laplacian on M.     

On string topology of three manifolds

Let M be a closed, oriented and smooth manifold of dimension d. Let LM be the space of smooth loops in M. In [String topology, preprint math.GT/9911159] Chas and Sullivan introduced the loop product, a product of degree -d on the homology of LM. We aim at identifying the three manifolds with “nontrivial” loop product. This is an application of some existing powerful tools in three-dimensional topology such as the prime decomposition, torus decomposition, Seifert fiber space theorem, torus theorem.     

Two properties of M-matrices

If A is an M-matrix with the property that some power of A is lower triangular, then A is lower triangular. An analogue of the Minkowski determinant theorem is proved for a subclass of the M-matrices.     

A Bernstein Theorem for the Abreu Equation

In this paper we prove a Bernstein type theorem for the Abreu equation on a complete Riemannian manifold (M,G ^*). Using this theorem and affine blow-up analysis we obtain interior estimates for the Abreu equation.     

A PλM-policy for an M/G/1 queueing system

We introduce P_λ^M-service policy for an M/G/1 queueing system. The stationary distribution of the workload under this policy is explicitly obtained through a decomposition technique, renewal reward theorem, and level crossing argument.