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We study Gaussian beams for the wave equation on a Riemannian manifold. For the transport equation we geometrize the leading term at the center of the Gaussian beam. More precisely, if is a Gaussian beam propagating along a geodesic c, then we show that where C is a constant and Y is a complex Jacobi tensor. Using a constant of motion for the non-linear Riccati equation related to the Jacobi equation, we prove that asymptotically the leading term of the energy carries constant energy.

We study Maxwell’s equations on a 4-manifold where the electromagnetic medium is modeled by an antisymmetric 2 2 -tensor with 21 real coefficients. In this setting the Fresnel surface is a fourth-order polynomial surface that describes the dynamical response of the medium in the geometric optics limit. For example, in an isotropic medium the Fresnel surface is a Lorentz null cone. The contribution of this paper is the pointwise description of all electromagnetic medium tensors κ with real coefficients that satisfy the following three conditions: (i) medium κ is invertible, (ii) medium κ is skewon-free, or non-dissipative, (iii) the Fresnel surface of κ is the union of two distinct Lorentz null cones. We show that there are only three classes of media with these properties and give explicit expressions in local coordinates for each class.

A Gaussian beam is an asymptotic solution to Maxwell’s equations that propagate along a curve; at each time instant its energy is concentrated around one point on the curve. Such a solution is of the form E = Re{eE0(x, t)}, where E0 is a complex vector field, P > 0 is a big constant, and θ is a complex second order polynomial in coordinates adapted to the curve. In recent work by A. P. Kachalov, electromagnetic Gaussian beams have been studied in a geometric setting. Under suitable conditions on the media, a Gaussian beam is determined by Riemann-Finsler geometry depending only on the media. For example, geodesics are admissible curves for Gaussian beams and a curvature equation determines the second order terms in θ. This work begins with a derivation of the geometric equations for Gaussian beams following the work of A. P. Kachalov. The novel feature of this work is that we characterize a class of inhomogeneous anisotropic media where the induced geometry is Riemannian. Namely, if e, μ are simultaneously diagonalizable with eigenvalues ei, μj , the induced geometry is Riemannian if and only if eiμj = ejμi for some i = j. What is more, if the latter condition is not met, the geometry is ill-behaved. It is neither smooth nor convex. We also calculate Riemannian metrics for different media. In isotropic media, gij = eμ δij and in more complicated media there are two Riemannian metrics due to different polarizations.

Suppose S is a semispray on a manifold M . We know that the complete lift S c of S is a semispray on T M with the property that geodesics of S c correspond to Jacobi fields of S . In this note we generalise this result and show how geodesic variations of k variables are related to geodesics of the k th iterated complete lift of S . Moreover, for sprays (that is, homogeneous semisprays) we show how geodesic variations of ( n − 1 ) variables are related to a natural generalisation of Jacobi tensors.

We study Maxwell's equations on a 4-manifold where the electromagnetic medium is described by an antisymmetric $2\choose 2$-tensor $\kappa$. In this setting, the Tamm-Rubilar tensor density determines a polynomial surface of fourth order in each cotangent space. This surface is called the Fresnel surface and acts as a generalisation of the light-cone determined by a Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speed as a function of direction. Favaro and Bergamin have recently proven that if $\kappa$ has only a principal part and if the Fresnel surface of $\kappa$ coincides with the light cone for a Lorentz metric $g$, then $\kappa$ is proportional to the Hodge star operator of $g$. That is, under additional assumptions, the Fresnel surface of $\kappa$ determines the conformal class of $\kappa$. The purpose of this paper is twofold. First, we provide a new proof of this result using Gr\"obner bases. Second, we describe a number of cases where the Fresnel surface does not determine the conformal class of the original $2\choose 2$-tensor $\kappa$. For example, if $\kappa$ is invertible we show that $\kappa$ and $\kappa^{-1}$ have the same Fresnel surfaces., Comment: 23 pages, 1 figure

To a system of second-order ordinary differential equations one can assign a canonical nonlinear connection that describes the geometry of the system. In this paper, we develop a geometric setting that also allows us to assign a canonical nonlinear connection to a system of higher-order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third- or fourth-order ordinary differential equations.

We use Frolicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is a Lagrangian vector field. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray.

Suppose $TM\setminus \{0\}$ and $T\widetilde M\setminus\{0\}$ are slashed tangent bundles of two smooth manifolds $M$ and $\widetilde M$, respectively. In this paper we characterize those diffeomorphisms $F\colon TM\setminus\{0\} \to T\widetilde M\setminus\{0\}$ that can be written as $F = (D\phi)|_{TM\setminus\{0\}}$ for a diffeomorphism $\phi\colon M\to \wt M$. When $F = (D\phi)|_{TM\setminus\{0\}}$ one say that $F$ \emph{descends}. If $M$ is equipped with two sprays, we use the characterization to derive sufficient conditions that imply that $F$ descends to a totally geodesic map. Specializing to Riemann geometry we also obtain sufficient conditions for $F$ to descent to an isometry.

In this paper, we define a complete lift for semisprays. If S is a semispray on a manifold M, its complete lift is a new semispray Scon TM. The motivation for this lift is two-fold: First, geodesics for Sccorrespond to the Jacobi fields for S, and second, this complete lift generalizes and unifies previously known complete lifts for Riemannian metrics, affine connections, and regular Lagrangians. When S is a spray, we prove that the projective geometry of Scuniquely determines S. We also study how symmetries and constants of motions for S lift into symmetries and constants of motions for Sc.

We study the wave equation on a bounded domain $M$ in $\mathbb{R}^m$ or on a compact Riemannian manifold with boundary. Assume that we do not know the coefficients of the wave equation but are given only the hyperbolic Robin-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. In this paper we show that at a fixed time $t_0$ a wave can be cut off outside a suitable set. That is, if $N\subset M$ is a union of balls in the travel time metric having centers at the boundary, then we can modify a given Robin boundary value of a wave such that at time $t_0$ the modified wave is arbitrarily close to the original wave inside $N$ and arbitrarily small outside $N$. Also, at time $t_0$ the time derivative of the modified wave is arbitrarily small in all of $M$. We apply this result to construct a sequence of Robin boundary values so that at a time $t_0$ the corresponding waves converge to a delta distribution $\delta_{\widehat{x}}$ while the time derivative of the waves converge to zero. Such boundary values are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to measured data and compute boundary values for the next iteration step. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the point $\widehat{x}$ where the wave focuses is known in travel time coordinates, and $\widehat{x}$ satisfies a certain geometrical condition.

We study Maxwell's equations on a 4-manifold N with a medium that is non-dissipative and has a linear and pointwise response. In this setting, the medium can be represented by a suitable (2,2)-tensor on the 4-manifold N. Moreover, in each cotangent space on N, the medium defines a Fresnel surface. Essentially, the Fresnel surface is a tensorial analogue of the dispersion equation that describes the response of the medium for signals in the geometric optics limit. For example, in isotropic medium the Fresnel surface is at each point a Lorentz light cone. In a recent paper, I. Lindell, A. Favaro and L. Bergamin introduced a condition that constrains the polarisation for plane waves. In this paper we show (under suitable assumptions) that a slight strengthening of this condition gives a pointwise characterisation of all medium tensors for which the Fresnel surface is the union of two distinct Lorentz null cones. This is for example the behaviour of uniaxial medium like calcite. Moreover, using the representation formulas from Lindell et al. we obtain a closed form representation formula that pointwise parameterises all medium tensors for which the Fresnel surface is the union of two distinct Lorentz null cones. Both the characterisation and the representation formula are tensorial and do not depend on local coordinates.

An area metric is a [Formula: see text]-tensor with certain symmetries on a 4-manifold that represents a non-dissipative linear electromagnetic medium. A recent result by Schuller, Witte and Wohlfarth gives a pointwise algebraic classification for such area metrics. This result is similar to the Jordan normal form theorem for [Formula: see text]-tensors, and the result shows that pointwise area metrics divide into 23 metaclasses and each metaclass requires two coordinate representations. For the first 7 metaclasses, we show that only one coordinate representation is needed. For the remaining 16 metaclasses we find an additional third coordinate representation.

We study the complex Riccati tensor equation D u cG + GCG − R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space. When time moves forward, Gaussian beams move along geodesics, and the Riccati equation determines the Hessian of the phase function for the Gaussian beam. The imaginary part of a solution G describes how a Gaussian beam decays in different directions of space. The main result of the present work is that the real part of G is the shape operator of the phase front for the Gaussian beam. This result generalizes a known result for the Riccati equation in R 3 . The idea of the proof is to express the Riccati equation in Fermi coordinates adapted to the underlying geodesic. In Euclidean geometry we also study when the phase front is contained in the area of influence, or light cone.