A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.     

关于SOL流形具有常平均曲率旋转曲面的注记

研究非欧流形SOL 空间上共形平均曲率方程的可解性,通过研究轮廓曲线对具有平均曲率的旋转曲面进行分类。当这些旋转曲面的平均曲率为给定函数时,计算出相应轮廓曲线的微分方程。通过求解这些微分方程,给出旋转函数是其上共形平均曲率的充分条件。     

Density results relative to the Dirichlet energy of mappings into a manifold

Let ?? be a smooth, compact, oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps u_k:Bⁿ → ?? with an equibounded Dirichlet integral give rise to elements of the space cart^2,1 (Bⁿ × ??). Assume that ?? is 1‐connected and that its 2‐homology group has no torsion. In any dimension n we prove that every element T in cart^2,1 (Bⁿ × ??) with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u_k:Bⁿ → ?? with Dirichlet energies converging to the energy of T. © 2006 Wiley Periodicals, Inc.     

A lower bound for the volume of a four-dimensional closed manifold

A closed orientable four-dimensional manifoldM ⁴, whose sectional curvatures satisfy –1 K 1, is considered. If the product of the curvatures of orthogonal area elements is nonnegative and if at least at one point all the curvatures are different from zero, then the estimate vol(M ⁴) > (4/9)² is obtained for the volume ofM ⁴. A theorem on the local structure of a manifold with small volume whose curvatures at every point are of the same sign is established.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 3–8, 1991.     

Comparison theorems for the volume of a complex submanifold of a Kaehler manifold

LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK _H≥4λ and antiholomorphic Ricci curvatureρ _A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P ⁿ⁻¹(λ) andM=C P ⁿ(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.     

The volume of a solid of revolution in calculus

<正>On the last time,you have known what a right circular cylinder is in calculus.Right circular cylinders are just a part of solids of revolution.Today you will learn solids of revolution arent right circular cylinders.In general way,a solid is bounded by the region under the curve y=f(x)by rotating a-     

The volume of a solid of revolution in calculus

On the last time, you have known what a right circular cylinder is in calculus. Right cir- cular cylinders are just a part of solids of revo- lution. Today you will learn solids of revolu- tion aren＇t right circular cylinders.     

On the volume of the projectivized tangent bundle in a complex Finsler manifold

We prove that the volume of PT _z ^{1, 0} M, calculated with respect to a Kähler metric induced by a complex Finsler structure, is a constant. This contrasts sharply with the situation in real Finsler geometry, where the volume of unit tangent sphere at each point x in a real Finsler manifold is in general a function of x. Furthermore, we point out that different metrics have different constants in general.     

Exponential iterated integrals and the relative solvable completion of the fundamental group of a manifold

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K.T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of π₁(M,x) can be expressed via these new integrals. The ring of exponential iterated integrals contains the coordinate rings for a class of universal representations, called the relative solvable completions of π₁(M,x). We consider exponential iterated integrals in the particular case of fibered knot complements, where the fundamental group always has a faithful relative solvable completion.     

Comparison theorems for the volume of a geodesic ball with a product of space forms as a model

We prove two comparison theorems for the volume of a geodesic ball in a Riemannian manifold taking as a model a geodesic ball in a product of two space forms.Work partially supported by the Fundació Caixa Castelló grant n. MI25043/92.Work partially supported by the DGICYT Grant No. PB91-0324.     

The Stasheff model of a simply-connected manifold and the string bracket

We revisit Stasheff's construction of a minimal Lie-Quillen model of a simply-connected closed manifold using the language of infinity-algebras. This model is then used to construct a graded Lie bracket on the equivariant homology of the free loop space of minus a point similar to the Chas-Sullivan string bracket.

     

Estimation of relative risk using a log-binomial model with constraints

The relative risk/prevalence ratio and odds ratio are very popular in medical research and epidemiological studies. The odds ratio is overused in practice due to its direct relation with the logistic regression. Interpreting the odds ratio in terms of “relative risk” may lead to incorrect inference on the prevalence of certain event. In this paper, we propose to estimate the relative risk using the log-binomial model by maximizing the likelihood with a linear constraint, which can be easily implemented by an existing function, such as “constrOptim” in R. Furthermore, we review other existing methods and compare their performance under a variety of settings using simulated data. We suggest the proposed and COPY methods in practice, based on its performance in the simulations. For illustration, we investigate the vigorous physical activity effects on obesity using data from the National Health and Nutrition Examination Survey, and the treatment effects on lung cancer mortality using data in R.     

Harmonic maps from a Riemannian manifold with a pole into an Hadamard manifold with negative sectional curvatures

In this paper we show a nonexistence result for harmonic maps with a rotational nondegeneracy condition from a Riemannian manifoldM with polep ₀ to a negatively curved Hadamard manifold under the condition that the metric tensor ofM is bounded and that the sectional curvature ofM at a pointp is bounded from below by −c dist(p ₀,p)⁻² (c: a positive constant) as dist(p ₀,p)→∞. Partly supported by Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan     

Global decomposition of a Lorentzian manifold as a Generalized Robertson-Walker space

Generalized Robertson-Walker (GRW) spaces constitute a quite important family in Lorentzian geometry, and it is an interesting question to know whether a Lorentzian manifold can be decomposed in such a way. It is well known that the existence of a suitable vector field guaranties the local decomposition of the manifold. In this paper, we give conditions on the curvature which ensure a global decomposition and apply them to several situations where local decomposition appears naturally. We also study the uniqueness question, obtaining that the de Sitter spaces are the only nontrivial complete Lorentzian manifolds with more than one GRW decomposition. Moreover, we show that the Friedmann Cosmological Models admit an unique GRW decomposition, even locally.     

The Ricci flow as a geodesic on the manifold of Riemannian metrics

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.