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Let (M, g) be a compact smooth N-dimensional Riemannian manifold without boundary. We consider the multiple existence of positive solutions of the problemwhere Δg stands for the Laplacian in M and f ε C2(M).

We present sufficient conditions for Anosov-type maps on Lie groups or Riemannian manifolds to have fixed points.

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

Matsumoto [10] remarked that some locally projectively flat Finsler spaces of non-zero constant curvature may be Riemannian spaces of non-zero constant curvature. The Riemannian connection, however, must be metric compatible, and this requirement places restrictions on the geodesic coefficients for the Finsler space in the form of a system of partial differential equations. In this paper, we derive this system of equations for the case where the geodesic coefficients are quadratic in the tangent space variables yi, and determine the solutions. We recover two standard Remannian metrics of non-zero constant curvature from this class of solutions.

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

We prove that a Riemannian metric on the 2-sphere or the projective plane can be C 2 approximated by a C ∞ metric whose geodesic flow has an elliptic closed geodesic.

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then given two points, there exists at least one piecewise smooth horizontal curve in this hypersurface connecting them. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces., 18 pages; some errors corrected

The best-constant problem for Nash and Sobolev inequalities on Riemannian manifolds has been intensively studied in the last few decades, especially in the compact case. We treat this problem here for a more general family of Gagliardo–Nirenberg inequalities including the Nash inequality and the limiting case of a particular logarithmic Sobolev inequality. From the latter, we deduce a sharp heat-kernel upper bound.AMS 2000 Mathematics subject classification: Primary 58J05

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K M ≥ 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than π/2 and contains a geodesic loop of length 2π. We also prove that if M is an n(≥3)-dimensional complete connected Riemannian manifold with K M ≥ 1 and radius bigger than π/2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.

We use Jacobi field arguments and the contraction mapping theorem to locate Fréchet means of a class of probability measures on locally symmetric Riemannian manifolds with non-negative sectional curvatures. This leads, in particular, to a method for estimating Fréchet mean shapes, with respect to the distance function ρ determined by the induced Riemannian metric, of a class of probability measures on Kendall's shape spaces. We then combine this with the technique of ‘horizontally lifting’ to the pre-shape spheres to obtain an algorithm for finding Fréchet mean shapes, with respect to ρ, of a class of probability measures on Kendall's shape spaces in terms of the vertices of random shapes. This gives us, for example, an algorithm for finding Fréchet mean shapes of samples of configurations on the plane which is expressed directly in terms of the vertices.

Using the bubbling argument of Sacks and Uhlenbeck, we prove the existence of n-harmonic maps from the n-sphere to Riemannian manifolds. An application is made to a problem concerning manifolds with strongly pth moment stable stochastic dynamical systems.

A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

Let S be an immersed compact surface into a Riemannian manifold M. We denote by H and K the mean curvature vector field of S and the sectional curvature function of M with respect to the tangent space of S and defineformula here

Our main result shows that a conformal mapping of hyperbolic n-space into another Riemannian manifold with scalar curvature bounded above by −n(n − 1) is necessarily distance decreasing. This is a generalisation of Ahlfors' version of the Schwarz-Pick lemma to Riemannian Geometry.

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.

We present a new proof of a strengthened version of Anosov's multiphase averaging theorem, originally stated for systems of ODEs with slow variables evolving in Rm and fast variables evolving on a smooth immersed manifold. Our result allows the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector field to have only Sobolev regularity. This is accomplished using normal form techniques adapted to a slightly generalized version of the DiPema-Lions theory of generalized flows for ODEs.

In previous papers [ 4 , 6 ], B.-Y. Chen introduced a Riemannian invariant δ M for a Riemannian n -manifold M n , namely take the scalar curvature and subtract at each point the smallest sectional curvature. He proved that every submanifold M n in a Riemannian space form R m (e) satisfies: δ M [les ][ n 2 ( n −2)]/ 2( n −1) H 2 +[half]( n +1)( n −2)e. In this paper, first we classify constant mean curvature hypersurfaces in a Riemannian space form which satisfy the equality case of the inequality. Next, by utilizing Jacobi's elliptic functions and theta function we obtain the complete classification of conformally flat hypersurfaces in Riemannian space forms which satisfy the equality.

We improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

We prove a Schwarz Lemma for conformal mappings between two complete Riemannian manifolds when the domain manifold has Ricci curvature bounded below in terms of its distance function. This gives a partial result to a conjecture of Chua.

For any ! > 0, we construct an explicit smooth Riemannian metric on the sphere S n , n " 3, that is within ! of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ! -dense in the unit tangent bundle. Moreover, for any ! > 0, we construct a smooth Riemannian metric on S n ,n " 3, that is within ! of the round metric and has a geodesic for which the complement of the closure of the correspondingorbit of the geodesic flow has Liouville measure less than ! .

In this note we use the Hopf map to construct a family of metrics in the 3-sphere parametrized on the space of positive smooth functions in the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. Also, the fibres are all geodesics if and only if the metric comes from a constant function and so, we have a Berger 3-sphere. Every geodesic in a 3-dimensional Riemannian manifold is a minimum for each elastic energy functional. Therefore, we characterize those func- tions on the 2-sphere that locally give metrics which have all the fibres being elastica, i.e., critical points of those functionals. Some applications are given including one to the Willmore-Chen variational problem.

We show that an isometric immersion of an n−dimensional compact Riemannian manifold of non-negative Ricci curvature with scalar curvature always less than n(n−1)λ−2 into a Euclidean space of dimension n + 1 can never be contained in a ball of radius λ.

Let (, g) be the standard Euclidean space or a Riemannian symmetric space of non-compact type of rank one. Let G be the identity component of the Lie group of all isometries of (, g). Let Γ be a discrete subgroup of G acting fixed point freely on whose quotient manifold MΓ is compact.

We are concerned with the value distribution of a mapping of an open Riemanniann-space (n≧ 3) into a Riemanniann-space. The value distribution theory of an analytic mapping of Riemann surfaces was initiated by S. S. Chern [1] and developed mainly by L. Sario [8], [9], [10], [11], and then by H. Wu [14], [15]. The most crucial part in Sario’s theory is the introduction of a kernel function on an arbitrary Riemann surface to describe appropriately the proximity of two points. His method indicates that the potential theoretic method is one of the powerful methods in the value distribution theory.

Let M be a smooth compact manifold and let be a smooth codimension q Riemannian foliation of M. Let T(M) be the tangent bundle of M and let E ⊂ T(M) be the subbundle tangent to . We may regard the normal bundle Q = T(M)/E of as a subbundle of T(M) satisfying T(M) = E ⊕ Q. Let g be a smooth Riemannian metric on Q invariant under the natural parallelism along the leaves of .

The generalised Whittaker vector is Λμ which is prevented from vanishing by rejection of the constancy of ω, previously assumed by all writers. It is shown that (1) the null divergence of Λμ is equivalent to Dirac's equation, (2) the length of Λμ measures the probability of occurrence of the electron (3) components of Λμ are connected with the Dirac wave functions and.possible transformations of x5 are probably related to the Uncertainty Principle.

Let M be an n-dimensional (n = 2m + 1, m ≦ 1) real differentiable manifold. if on M there exist a tensor field , a contravariant vector field ξi and a convariant vector field ηi such that then M is said to have an almost contact structure with the structure tensors (φ,ξ, η) [1], [2]. Further, if a positive definite Riemannian metric g satisfies the conditions then g is called an associated Riemannian metric to the almost contact structure and M is then said to have an almost contact metric structure. On the other hand, M is said to have a contact structure [2], [4] if there exists a 1-form η over M such that η ∧ (dη)m ≠ 0 everywhere over M where dη means the exterior derivation of η and the symbol ∧ means the exterior multiplication. In this case M is said to be a contact manifold with contact form η. It is known [2, Th. 3,1] that if η = ηidxi is a 1-form defining a contact structure, then there exists a positive definite Riemannian metric in gij such that and define an almost contact metric structure with and ηi where the symbol ∂i standing for ∂/∂xi.

1. The classification of Riemann surfaces with respect to the equation Δu = Pu (P ≥ 0, P ≢ 0) was initiated by Ozawa [13] and further developed by L. Myrberg [8, 9], Royden [14], Nakai [10, 11], Sario-Nakai [15], Nakai-Sario [12], Glasner-Katz [3], and Kwon-Sario [7].

Let (M,g) be a Riemannian manifold and let R be its Riemannian curvature tensor. If (M, g) is a locally symmetric space, we have(*) R(X,Y)·R = 0 for all tangent vectors X,Ywhere the endomorphism R(X,Y) (i.e., the curvature transformation) operates on R as a derivation of the tensor algebra at each point of M. There is a question: Under what additional condition does this algebraic condition (*) on R imply that (M,g) is locally symmetric (i.e., ∇R = 0)? A conjecture by K. Nomizu [5] is as follows : (*) implies ∇R = 0 in the case where (M, g) is complete and irreducible, and dim M ≥ 3. He gave an affirmative answer in the case where (M,g) is a certain complete hypersurface in a Euclidean space ([5]).