Higher covariant derivatives

We obtain a composition formula for the higher covariant derivatives on a vector bundle over a manifold. The formula generalizes the classical Leibniz product rule for the derivative. We also obtain, as a corollary, a generalization of the author’s theorem about the double exponential map to the case of multiple maps.     

Exponential maps of Sobolev metrics on loop groups

We find conditions for the exponential map on a weak riemannian Hilbert manifold to be a nonlinear Fredholm map of index zero and apply the result to left-invariant Sobolev metrics on loop groups.

     

On the basic algebraic operations in solvable Lie groups

Summary For a simply connected solvable Lie group we specify the structure of the product, the inverse and the exponential map expressed in suitable coordinates (canonical coordinates of the second kind), and point out that in these coordinates the product and inverse are expressed entirely in terms of polynomials, exponential functions and trigonometric functions. We devise algorithms for computing the product, the inverse and the exponential map.     

Random methods in 3-manifold theory

The surface map arising from a random walk on the mapping class group may be used as the gluing map for a Heegaard splitting, and the resulting 3-manifold is known as a random Heegaard splitting. We show that the splitting distance of random Heegaard splittings grows linearly in the length of the random walk, with an exponential decay estimate for the proportion with slower growth. We use this to obtain the limiting distribution of Casson invariants of random Heegaard splittings.     

Geodesic flow on the diffeomorphism group of the circle

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.     

Transversal wave maps and transversal exponential wave maps

Transversal wave maps and wave maps are different. There are wave maps which are not transversal wave maps, and vice versa. However, if f is a wave map under certain circumstance, then f is a transversal wave map. We show that if f is a transversal exponential wave map, then the associated energy–momentum is transversally conserved. We finally obtain the relationship among transversal wave maps, transversal exponential wave maps and certain second order symmetric tensors.     

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.     

The Exponential Map on the Free Loop Space is Fredholm

We study the exponential map on Hilbert riemannian manifolds and show that on the free loop space it is a nonlinear Fredholm map of index zero. Among corollaries we obtain that only one type of conjugate points exists in . This answers a question of W. Klingenberg. Submitted: December 1996, revised version: March 1997.     

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincaré maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.     

Aspects of the theoretical and numerical formulation of single crystal plasticity at finite strains

We give an overview about an incremental variational formulation of single crystal plasticity at finite strains and discuss an approach to the computation of the exponential map and its derivatives based on a spectral representation.     

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

Prime graphs and exponential composition of species

In this paper, we enumerate prime graphs with respect to the Cartesian multiplication of graphs. We use the unique factorization of a connected graph into the product of prime graphs given by Sabidussi to find explicit formulas for labeled and unlabeled prime graphs. In the case of species, we construct the exponential composition of species based on the arithmetic product of species of Maia and Méndez, and express the species of connected graphs as the exponential composition of the species of prime graphs.     

The Leibniz formula for the covariant derivative and some of its applications

We obtain a formula for the higher covariant derivatives on the tensor product of vector bundles which is a wide generalization of the classical Leibniz formula. We construct an algorithm for the calculation of the part of the Taylor series of the double exponential map linear with respect to the second variable.     

The affine connection in the normal coordinates

We propose a method to calculate the derivatives of the Christoffel symbols in the normal coordinates on a smooth manifold with symmetric affine connection. As an application, we construct an algorithm for computing the Taylor series of the double exponential map.     

Large deviations bound for semiflows over a non-uniformly expanding base

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical set of the base map. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the semiflow, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast as time goes to infinity. The arguments need the base transformation to exhibit exponential slow recurrence to the singular set which, in all known examples, implies exponential decay of correlations. Suspension semiflows model the dynamics of flows admitting cross-sections, where the dynamics of the base is given by the Poincaré return map and the roof function is the return time to the cross-section. The results are applicable in particular to semiflows modeling the geometric Lorenz attractors and the Lorenz flow, as well as other semiflows with multidimensional non-uniformly expanding base with non-flat singularities and/or criticalities under slow recurrence rate conditions to this singular/critical set. We are also able to obtain exponentially fast escape rates from subsets without full measure. *The author was partially supported by CNPq-Brazil and FCT-Portugal through CMUP and POCI/MAT/61237/2004.     

Schanuel property for additive power series

We prove a version of Schanuel’s Conjecture for a field of Laurent power series in positive characteristic replacing ? and a non-algebraic additive power series replacing the exponential map.     

The equator map and the negative exponential functional

We define a negative exponential harmonic map from the ballB ⁿ of ℝ_n into the sphereS ⁿ of ℝ ⁿ⁺¹ . We prove that the equator map is a negative exponential harmonic map, but not stable for the negative exponential functional whenn≥2. Moreover, we consider maps from a ballB ⁿ into the unit sphereS ^m of ℝ_m+1 wherem≥2, and prove that no nonconstant, non surjective map can reach either the minimum or the maximum of the negative exponential functional.     

Radial solutions for a prescribed mean curvature equation with exponential nonlinearity

We establish an existence result for radial solutions for a prescribed mean curvature equation with exponential nonlinearity. Our methods are based on degree theory combined with a time map analysis. We also obtain two nonexistence results for positive solutions for more general f; one of them is not limited to radial solutions.     

Conformal normal coordinates

We show that in every class of conformally equivalent Riemannian spaces there exists at least one space whose exponential map is volume preserving near a given point. The proof is based on the solution of degenerated non-linear elliptic equations within the framework of weighted Hölder spaces.     

Piecewise interpolants on matrix lie groups

Here we consider a numerical procedure to interpolate on matrix Lie groups. By using the exponential map and its (1, 1) diagonal Padé approximant, piecewice interpolants may be derived. The approach based on the Padé map has the advantage that the computation of exponentials and logarithms of matrices are reduced. We show that the updating technique proposed by Enright in [1] may be applied when a dense output is required. The application to the numerical solution of a system ODEs on matrix group and to a classical interpolation problem are reported.