# Ioan BucataruUniversitatea Alexandru Ioan Cuza | UAIC · Department of Mathematics

Ioan Bucataru

Ph. D.

## About

73

Publications

9,992

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

917

Citations

Introduction

Additional affiliations

Education

April 1995 - March 1998

## Publications

Publications (73)

Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we...

We prove that in a Finsler manifold with vanishing χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document}-curvature (in particular with constant flag cu...

We prove that in a Finsler manifold with vanishing $\chi$-curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald c...

Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we...

We use two non-Riemannian curvature tensors, the χ-curvature and the mean Berwald curvature to characterise a class of Finsler metrics admitting first integrals. This class includes Finsler metrics of constant flag curvature.

We use two non-Riemannian curvature tensors, the $\chi$-curvature and the mean Berwald curvature to characterise a class of Finsler metrics admitting first integrals.

We present a new proof of a Finslerian version of Beltrami's theorem (1865) which works also in dimension 2.

We present a new proof of a Finslerian version of Beltrami's theorem
(1865) which works also in dimension 2.

In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic identity appears as an obstruction to the formal integrability of some operators in Finsler geometry, [4,7]. Thi...

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classical projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature...

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature,...

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous...

In this paper we study the invariant metrizability and projective
metrizability problems for the special case of the geodesic flow associated to
the canonical connection of a Lie group. We prove that the canonical connection
is projectively Finsler metrizable if and only if it is Riemann metrizable.
This result means that the structure is rigid in...

In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature tensor, [14, page 184]. In this paper, we provide an answer to this question, within the class of Finsler metrics of scalar flag curvature. In Theorem 3.1, we show that the answer is
negative, for non-vanishing scalar flag...

In his book "Differential Geometry of Spray and Finsler spaces", page 177, Zhongmin Shen asks ``wether or not there always exist non-trivial Funk functions on a spray space''. In this note, we will prove that the answer is negative for the geodesic spray of a Finslerian function of non-vanishing scalar flag curvature.

In this paper we provide generalized Helmholtz conditions, in terms of a
semi-basic 1-form, which characterize when a given system of second order
ordinary differential equations is equivalent to the Lagrange equations, for
some given arbitrary non-conservative forces. For the particular cases of
dissipative or gyroscopic forces, these conditions,...

By generalizing the cosymplectic setting for time-dependent Lagrangian
mechanics, we propose a geometric framework for the Lagrangian formulation of
classical field theories with a Lagrangian depending on the independent
variables. For that purpose we consider the first order jet bundles $J^1\pi$ of
a fiber bundle $\pi:E\to {\mathbb R}^k$ where ${\...

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In our main result we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are me...

We use the Fr\"olicher-Nijenhuis formalism to reformulate the inverse problem
of the calculus of variations for a system of differential equations of order
2k in terms of a semi-basic 1-form of order k. Within this general context, we
use the homogeneity proposed by Crampin and Saunders in [14] to formulate and
discuss the projective metrizability...

In this paper we characterize sprays that are metrizable by Finsler functions
of constant flag curvature. By solving a particular case of the Finsler
metrizability problem we provide the necessary and sufficient conditions that
can be used to decide whether or not a given homogeneous system of second order
ordinary differential equations represents...

In this paper we study symmetries, Newtonoid vector fields, conservation
laws, Noether's Theorem and its converse, in the framework of the
$k$-symplectic formalism, using the Fr\"olicher-Nijenhuis formalism on the
space of $k^1$-velocities of the configuration manifold.
For the case $k=1$, it is well known that Cartan symmetries induce and are
indu...

Suppose $S$ is a semispray on a manifold $M$. We know that the complete lift
$S^c$ of $S$ is a semispray on $TM$ with the property that geodesics of $S^c$
correspond to Jacobi fields of $S$. In this note we generalize this result and
show how geodesic variations of $k$-variables are related to geodesics of the
$k$th iterated complete lift of $S$. M...

In this work we show that for the geodesic spray $S$ of a Finsler function
$F$ the most natural projective deformation $\widetilde{S}=S -2 \lambda
F\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of
$\lambda \in \mathbb R$. This result shows how rigid is the metrizablility
property with respect to certain reparameterizati...

A geometric setting for studying higher order ordinary differ-ential equations (HODE) is obtained by choosing a nonlinear connection associated to the HODE. One such nonlinear connection was introduced in local coordinates by Miron and Atanasiu [23]. In this note we summarize results from [8], and show that this nonlinear connection has many of the...

The projective metrizability problem can be formulated as follows: under what
conditions the geodesics of a given spray coincide with the geodesics of some
Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective
metrizability problem for a spray in terms of a first-order partial
differential operator $P_1$ and a set of algeb...

To a system of second order ordinary differential equations (SODE) one can
assign a canonical nonlinear connection that describes the geometry of the
system. In this work we develop a geometric setting that allows us to assign a
canonical nonlinear connection also to a system of higher order ordinary
differential equations (HODE). For this nonlinea...

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\col...

We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Fr\"olicher-Nijenhuis theory on the first jet bundle, $J^1\pi$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian if and only if a special cl...

From a spray space $S$ on a manifold $M$ we construct a new geometric space $P$ of larger dimension with the following properties: 1. Geodesics in $P$ are in one-to-one correspondence with parallel Jacobi fields of $M$. 2. $P$ is complete if and only if $S$ is complete. 3. If two geodesics in $P$ meet at one point, the geodesics coincide on their c...

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 $S^c$ on $TM$. The motivation for this lift is two-fold: First, geodesics for $S^c$ correspond to the Jacobi fields for $S$, and second, this complete lift generalizes and unifies previously known complete lifts for...

A system of third order differential equations, whose coefficients do not depend explicitly on time, can be viewed as a third order vector field, which is called a semispray, and lives on the second order tangent bundle. We prove that a regular second order Lagrangian induces such a semispray, which is uniquely determined by two associated Poincaré...

We have proved that the innermost wavefront-slowness sheet of a Hookean solid is convex, whether or not it is detached from the other sheets. This theorem is valid for the generally anisotropic case, and it is an extension of theorems whose proofs require the detachment of the innermost sheet. Although the Hookean solids that represent most materia...

We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is locally Lagrangian. 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 (...

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor, hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogo...

We construct an eighteen-dimensional orbifold that is in a one-to-one correspondence with the space of SO (3)-orbits of elasticity tensors. This allows us to obtain a local parametrization of SO (3)-orbits of elasticity tensors by six SO (6)-invariant and twelve SO (3)-invariant parameters. This process unravels the structure of the space of the or...

We construct a method for finding the elasticity parameters of an anisotropic homogeneous medium using only ray velocities and corresponding polarizations. We use a linear relation between the ray velocities and wavefront slownesses, which depends on the corresponding polarizations. Notably, this linear relation circumvents the need to use explicit...

For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange metric. For the particular case whe...

We show that the canonical semispray of a regular Lagrangian of order k is uniquely determined by two associated Cartan-Poincaré one-forms. Equivalently, the canonical semispray is uniquely determined by its canonical presymplectic structure and one of the Cartan-Poincaré one-forms. We prove that this k + 1 order vector field is determined by a var...

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a priory knowledge of the symmetry axes.

We show that, in general, wavefronts are more symmetric than the medium in which they propagate. This means that we cannot determine the symmetries of the medium based solely on the symmetries of the wavefronts. However, we show that we can determine the symmetries of the medium from the symmetries of wavefronts and polarizations together.

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these
conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a
priory knowledge of the symmetry axes.

The geometry of a nonconservative mechanical system is determined by its associated semispray and the corresponding nonlinear connection. The semispray is uniquely determined by the symplectic structure and the energy of the corresponding Lagrange space and the external force field. We prove that the corresponding nonlinear connection is uniquely d...

The geometry of a Lagrangian mechanical system is determined by its associated evolution semispray. We uniquely determine this semispray using the symplectic structure and the energy of the Lagrange space and the external force field. We study the variation of the energy and Lagrangian functions along the evolution and the horizontal curves and giv...

A material body with smoothly distributed microstructure can be seen geometrically as a fibre bundle. Within this very general
framework, we show that a theory of continuous distributions of dislocations can be formulated and specialized to particular
applications, both old and new.

We prove that in elastic anisotropic inhomogeneous media, rays and wavefronts are orthogonal to each other with respect to the metric induced by the phase-velocity function. The standard orthogonality of rays and wavefronts in elastic isotropic inhomogeneous media is a special case of this formulation.

For a system of (k+l) order differential equations (or a semispray of order k on the tangent bundle of order k) we determine a nonlinear connection induced by it. This nonlinear connection induces a linear connection D on the total space of the tangent bundle of order k, that is called the Berwald connection. Using the Cartan's structure equations...

Consider L a regular Lagrangian, S the canonical semispray, and h the horizontal projector of the canonical nonlinear connection. We prove that if the Lagrangian is constant along the integral curves of the Euler-Lagrange equa- tions then it is constant along the horizontal curves of the canonical nonlinear connection. In other words S(L) = 0 impli...

We prove that there are eight subgroups of the orthogonal group O(3) that determine all symmetry classes of an elasticity tensor. Then, we provide the necessary and sufficient conditions that allow us to determine the symmetry class to which a given elasticity tensor belongs. We also give a method to determine the natural coordinate system for each...

A material body with smoothly distributed microstructure can be seen geometrically as a fibration or, when the symmetry group is specified, as a fiber bundle. Within this very general framework, we present a geometric description of such material bodies in terms of fiber jets. We introduce the notion of fiber frame and construct the corresponding L...

A symmetry class of an elasticity tensor, c, is determined by the variance of this tensor with respect to a subgroup of the special orthogonal group, SO(3). Using the double covering of SO(3) by the special unitary group, SU(2), we determine the subgroups of SU(2) that correspond to each of the eight symmetry classes. A family of maps between C2 an...

After a review of 2- and 3-dimensional Finsler spaces from the Berwald and Cartan connections points-of-view, several tensors are explicitly worked out by means of Computer Algebra, and Moor frames are also used to obtain information about the almost flat metric
ds(2) = dr(2) + r(2)dOmega(2) - dt(2) + epsilondOmegadt,
derived from the Rutz's 2...

A material body with smoothly distributed microstructure can be seen geometrically as a fibre bundle. Within this very general framework, we show that a theory of continuous distributions of dislocations can be formulated and specialized to particular applications, both old and new.

Nonholonomic Holland frames are determined for Beil metrics and (α, β)-metrics using deformation theory and previous work.

We determine a nonholonomic Finsler frame for a class of Generalized La- grange spaces, for a class of Lagrange spaces with (ã;å)-metric and for Finsler spaces with (ã;å)-metric. Then, a special Finsler connection induced by such a nonholonomic frame is determined. Finally we study the integrability conditions for Cartan's structure equations of a...

The two types of KCC-theory are presented in detail, one without time in coefficients( type A) and one with time in the ciefficients(type B). Detailed examples from ecology are given.

It has been proved that the class of C-reducible Finsler spaces, intro-duced by M. Matsumoto, [11] reduces to Randers and Kropina spaces, [12]. In two previous paper, [2], [3], we have determined a canonical Finsler frame for these two Finsler spaces: Randers and Kropina. As is well known, these Finsler spaces have two canonical metrics, a Riemanni...

From a rigorous Finsler geometric perspective, we re-examine Hol-land's Randers space theory of motion of an electron in flat Minkowski space-time permeated with an electromagnetic field. Holland's theory was moti-vated by analogy with plastic deformation and dislocation in Bravais crystals, through work of D. Bohm on Quantum Mechanics. We develop...

The KCC-theory (Kosambi, [12], CARTAN, [7], and CHERN, [8]) of a system of second order ordinary differential equations (SODE) uses five geometric in-variants that determine, up to a change of coordinates, the solutions of the system. Geometrically speaking to a SODE corresponds a vector field, called a semispray (or alternatively , a second order...

First, using the complete lift of a linear connection we construct the horizontal lift of a vector field with the aid of an arbitrary semispray S. It is proved that this horizontal lift is independent on the choice of the semispray S. This reformulates well-known constructions for the case of tangent bundle, [4], [8].
Secondly, the complete and h...

The first part of this work is a natural extension of a recent paper by
M.Anastasiei ([1]). In §1 the Finsler connections are defined by local components.
In §2 a Finsler connection appears as a pair (N,∇), where N is a
nonlinear connection on the jet bundle of order k, OsckM and ∇ is a linear
connection in the pull-back bundle of the tangent bundl...

In the case of the tangent bundle, an almost product structure, for which the eigenspace corresponding to the eigenvalue -1 is the vertical subspace, can be expressed in the natural basis with a set of functions which are the local coefficients of a nonlinear connection. For a spray S on the tangent manifold TM, a nonlinear connection is associated...

We consider the first variation of those curves on tangent manifold T M which have property that are parallel with respect to the canoni-cal metrical connection in a generalised Lagrange space. Accordingly we introduce and study the Jacobi fields on T M. Several particular cases are discussed.

In the paper [8] a nonlinear connection on k-osculator bundle is charac-terised by a system of functions defined on each domain of local chart, which verify a special formula. Starting with this result, the present author associated in [2], to a nonlinear connection on the k-osculator bundle a special map, called connection map. The aim of this pap...

The geometry of the k-osculator bundle over a smooth manifold
M was developed by R.Miron and his school. It was used for the
geometrization of the higher order Lagrangians and the prolongation
of the Riemannian, Finslerian and Lagrangian structures, ([?]).
In this work we show that the prolongation of a Riemannian metric
provides a Riemannian subme...

Recently, R.Miron and Gh.Atanasiu ([4], [5], [6]) have developed a ge-ometrisation of the higher order Lagrangians which parallels that of the first order Lagrangians expounded in a monograph by R.Miron and M. Anastasiei ([3]). In this geometrization a central role is played by certain nonlinear connections in the k-osculator bundle. Our aim is to...

## Questions

Question (1)

I will appreciate very much if anybody knows about an English translation of Berwald's paper: Berwald, Ludwig; Über die n-dimensionalen Geometrien konstanter Krümmung, in denen die Geraden die kürzesten sind. (German) Math. Z. 30 (1929), no. 1, 449–469.

Many thanks, IB.

## Projects

Project (1)