We formulate a statistical analogy of regular Lagrange mechanics and Finsler geometry derived from Grisha Perelman's functionals generalized for nonholonomic Ricci flows. There are elaborated explicit constructions when nonholonomically constrained flows of Riemann metrics result in Finsler like configurations, and inversely, and geometric mechanics is modelled on Riemann spaces with preferred nonholonomic frame structure.
In this paper, it is elaborated the theory the Ricci flows for manifolds enabled with nonintegrable (nonholonomic) distributions defining nonlinear connection structures. Such manifolds provide a unified geometric arena for nonholonomic Riemannian spaces, Lagrange mechanics, Finsler geometry, and various models of gravity (the Einstein theory and string, or gauge, generalizations). We follow the method of nonholonomic frames with associated nonlinear connection structure and define certain classes of nonholonomic constraints on Riemann manifolds for which various types of generalized Finsler g…
We elaborate an unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructio…
We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart-Moffat and Finsler-Lagrange spaces with connections compatible to a general nonsymmetric metric structure. Elaborating a metrization procedure for arbitrary distinguished connections, we define the class of distinguished line…
In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost Kähler - Ricci geometric evolution equations. The se…
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduced $hh$-curvature on finite time. Next, it is shown that the evolution of Ricci scalar is a parabolic-type equation and if the initial Finsler metric is of positive flag curvature, then the flag curvature and the Ricci scalar remain positive as long as the solution exists. Finall…
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, existence of solutions to the so called Hamilton Ricci flow on Finsler spaces is studied and a short time solution is found. To this end the Finslerian Ricci-DeTurck flow on Finsler spaces is defined and existence of its solution in short time is proved. Next, this solution is pulled back to determine a short time solution to the Hamilton Ricci flow on underlying Finsler space.
In this work, convergence of evolving Finslerian metrics first in a general flow next under Finslerian Ricci flow is studied. More intuitively it is proved that a family of Finslerian metrics $g(t)$ which are solutions to the Finslerian Ricci flow converge in $C^{\infty}$ to a smooth limit Finslerian metric as $ t $ approaches the finite time $ T $. As a consequence of this result one can show that in a compact Finsler manifold the curvature tensor along Ricci flow blows up in short time.
Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group and hence the first de Rham cohomology group vanishes.
In this paper we have studied the class of Finsler metrics, called C3-like metrics which satisfy the un-normal and normal Ricci flow equation and proved that such metrics are Einstein.
Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to the original Ricci flow by pulling back the solution of the Ricci-DeTurck flow using appropriate diffeomorphisms. In the end, we illustrate this argument with some examples.
We use the Frölicher-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 problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projectivpre-e metrizability in terms of homogeneous semi-basic 1-forms. Such a semi-basic 1-form is the Poincaré-Cartan 1-form of a …
The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are given and some recent results are presented. They are a generalization of Einstein metrics and are previously developed by the present authors for Finsler manifolds. In the present work, it is shown that a complete shrinking Ricci soliton Finsler manifold has a finite fundamental group.
Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.
The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In this paper, we consider homogeneous Finsler spaces with square metric and Randers change of square metric. First, we derive the explicit formulae for Ricci curvature of homogeneous Finsler spaces with these metrics. Next, we find a necessary and sufficient condition under which a homogeneous Finsler space with either of these metrics is of vanishing $S$-curvature. The formulae for Ricci curvature of homogeneous Finsler spaces with square metric an…
In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative ge…
In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern connection, $\mathbf{S}-$curvature and Hessian) and suitable Ricci curvature bounds throughout the flow. One of the key tools we use is the Bochner identity for Finsler structures proved by Ohta and Sturm.
Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.
Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces $G/SO(n)$ provided with an N-connection structure and an adap…
The pullback approach to global Finsler geometry is adopted. Some new types of special Finsler spaces are introduced and investigated, namely, Ricci, generalized Ricci, projectively recurrent and m-projectively recurrent Finsler spaces. The properties of these special Finsler spaces are studied and the relations between them are singled out.
In this paper we introduce a natural definition for the affine maps between two Finsler manifolds $(M, F)$ and $(N,\tilde F)$ and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;N)$, in the sense of a Berwald nonlinear connection $Γ^b_jet$, a Berwald $Γ^b_jet$-linear d-connection $BΓ^b_jet$, together with its d-torsions and d-curvatures, which geometrically characterizes the initial affine maps between Finsler manifolds.
In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.
We establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop-Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet-Myers type on Finsler manifolds. Finally, we obtain a sharp Poincaré-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds.
The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces. Quasi-Einstein metrics serve also as solution to the Ricci flow equation. Here, the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is defined. In compact case, it is proved that the quasi-Einstein metrics are solution to the Finslerian Ricci flow and conversely, certain form of solutions to the Finslerian Ricci flow are quasi-Einstein Finsler metrics.
The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry and, in our approach, with fractional derivatives. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a "prime" Lagrange (in particular, F…
The book contains a collection of works on Riemann-Cartan and metric-affine manifolds provided with nonlinear connection structure and on generalized Finsler-Lagrange and Cartan-Hamilton geometries and Clifford structures modelled on such manifolds. The choice of material presented has evolved from various applications in modern gravity and geometric mechanics and certain generalizations to noncommutative Riemann-Finsler geometry. The authors develop and use the method of anholonomic frames with associated nonlinear connection structure and apply it to a number of concrete problems: constructi…
In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial differential system on the energy function can be reduced to a first order system on this same function. In this way we are able to give effective necessary and sufficient conditions for the local existence of a such Finsler metric in terms of the holonomy algebra generated by horizontal vector-fields. We also consider the Landsberg metrizability problem and prove sim…
The physics of classical particles in a Lorentz-breaking spacetime has numerous features resembling the properties of Finsler geometry. In particular, the Lagrange function plays a role similar to that of a Finsler structure function. A summary is presented of recent results, including new calculable Finsler structures based on Lagrange functions appearing in the Lorentz-violation framework known as the Standard-Model Extension.
Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our r…
We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.
A projective parameter of a geodesic on a Finsler space is defined to be solution of a certain ODE. Using projective parameter and Funk metric, one can construct a projectively invariant intrinsic pseudo-distance on a Finsler space. In the present work, solutions of the projective parameter's ODE are characterized with respect to the sign of parallel Ricci tensor of a Finsler space. It is shown that the pseudo-distance is trivial on complete Finsler spaces of positive semi-definite Ricci tensor and it is a distance on Finsler spaces of parallel negative definite Ricci tensor. These results gen…
The standard Bonnet-Myers theorem says that if the Ricci scalar of a Riemannian manifold is bounded below by a positive number, then the manifold is compact. Moreover, a bound of its diameter is pointed out. The theorem was extended to Finsler manifolds. In this paper we prove that if a certain condition on the average of the Ricci scalar holds, then the Finsler manifold M is compact if the Ricci scalar is bounded above by the same positive number. An upper bound of the diameter is also found. With no condition on Ricci scalar itself but with a different one on its average, we find that the Fi…
In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.
We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metric structure (with an additional affine connection possessing nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler, geometry modeled on nonholonomic manifolds. This results in the problem of constructing noncommutative geometries with local anisotropy, in parti…
We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler-Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function.
We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern-Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi-Civita connection of some Riemannian metric. We show for the first time that this result does not extend to Finsler spacetimes. More precisely, we find a large class of Berwald spacet…
The work extends the A. Connes' noncommutative geometry to spaces with generic local anisotropy. We apply the E. Cartan's anholonomic frame approach to geometrical models and physical theories and develop the nonlinear connection formalism for projective module spaces. Examples of noncommutative generation of anholonomic Riemann, Finsler and Lagrange spaces are analyzed. We also present a research on noncommutative Finsler--gauge theories, generalized Finsler gravity and anholonomic (pseudo) Riemann geometry which appear naturally if anholonomic frames (vierbeins) are defined in the context of…
The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems [S. Vacaru: J. Math. Phys. \textbf{49} (2008) 043504 \& Rep. Math. Phys. \textbf{63} (2009) 95] is extended to include geometric mechanics and gravity models on Lie algebroids. We prove that such evolution scenarios of geometric mechanics and analogous gravity can be modelled as gradient flows characterized by generalized Perelman functionals if an equivalent geometrization of Lagrange mechanics [J. Kern, Arch. Math. (Basel) \textbf{25} (1974) 438] is considered. The R. Hamilton equations on Lie algebr…
The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-Singer index formulas. There are considered certain applications in modern gravity and geometric mechanics of such Clifford-Lagrange/ Finsler gerbes and their realizations as nonholonomic Clifford and Riemann-Cartan modules.
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples, and find the cohomological obstractions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, wh…
Weighted Ricci curvatures with various weights of the S-curvature have played an important role in Finsler geometry, especially in the control of volumes of Finsler manifolds. In this paper, we study general weighted Ricci curvatures, and characterize Randers metrics of almost isotropic weighted Ricci curvatures.
The geometric constructions are elaborated on (semi) Riemannian manifolds and vector bundles provided with nonintegrable distributions defining nonlinear connection structures induced canonically by metric tensors. Such spaces are called nonholonomic manifolds and described by two equivalent linear connections also induced in unique forms by a metric tensor (the Levi Civita and the canonical distinguished connection, d-connection). The lifts of geometric objects on tangent bundles are performed for certain classes of d-connections and frame transforms when the Riemann tensor is parametrized by…
We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz-Finsler version of the Bonnet-Myers theorem based on a generalized Bishop inequality.
We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with $ε$-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie-Yeroshkin and Kuwae-L…
We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below. For the proof we combine techniques developed in these papers. Because the Finsler product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again …
We study the variational problem for $N$-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler-Lagrange equations for this type of variational problem.
We develop the method of anholonomic frames with associated nonlinear connection (in brief, N--connection) structure and show explicitly how geometries with local anisotropy (various type of Finsler--Lagrange--Cartan--Hamilton geometry) can be modeled in the metric--affine spaces. There are formulated the criteria when such generalized Finsler metrics are effectively induced in the Einstein, teleparallel, Riemann--Cartan and metric--affine gravity. We argue that every generic off--diagonal metric (which can not be diagonalized by coordinate transforms) is related to specific N--connection conf…
Operators on nonlinear metric measure spaces I: A New Laplacian comparison theorem on Finsler manifolds and a generalized Li-Yau approach to gradient estimates of Finslerian Schrödinger equation
We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution (Hamilton's) equations. We apply in fractional calculus the nonlinear connection formalism originally elaborated in Finsler geometry and generalizations and recently applied to classical and quantum gravity theories. There are also analyzed the fractional operators for the entrop…
