Generalized Berwald spaces are the Finsler spaces which admit metric linear connections in the ta... more Generalized Berwald spaces are the Finsler spaces which admit metric linear connections in the tangent bundle of their base manifold. They reduce to Berwald spaces if the torsion of this connection vanishes. We show that generalized Berwald spaces with base manifold R n are exactly the affine deformations of Minkowski spaces. They can also be represented as a pair of a Riemannian and a Minkowski space, and they coincide with the Finsler spaces of 1-form metric. We investigate their reduction to Minkowski, Riemannian or Euclidean spaces, as well as their conformal relations.
In this paper, we give an equivalent characterization of conformal vector fields on a Finsler man... more In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold [Formula: see text], whose metric [Formula: see text] is defined by a Riemannian metric [Formula: see text] and a 1-form [Formula: see text]. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds [Formula: see text] when [Formula: see text] is of constant sectional curvature and [Formula: see text] is a conformal 1-form with respect to [Formula: see text].
After a careful study of the mixed curvatures of the Berwald-type (in particular, Berwald) connec... more After a careful study of the mixed curvatures of the Berwald-type (in particular, Berwald) connections, we present an axiomatic description of the so-called Yano-type Finsler connections. Using the Yano connection, we derive an intrinsic expression of Douglas' famous projective curvature tensor and we also represent it in terms of the Berwald connection. Utilizing a clever observation of Z. Shen, we show in a coordinate-free manner that a "spray manifold" is projectively equivalent to an affinely connected manifold iff its Douglas tensor vanishes. From this result we infer immediately that the vanishing of the Douglas tensor implies that the projective Weyl tensor of the Berwald connection "depends only on the position".
We prove that a dieomorphism of a manifold with an Ehresmann connection is an automorphism of the... more We prove that a dieomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesics, considered as parametrized curves, to geodesics) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative).
In this survey we approach some aspects of tangent bundle geometry from a new viewpoint. After an... more In this survey we approach some aspects of tangent bundle geometry from a new viewpoint. After an outline of our main tools, i.e., the pull-back bundle formalism, we give an overview of Ehresmann connections and covariant derivatives in the pull-back bundle of a tangent bundle over itself. Then we define and characterize some special classes of generalized metrics. By a generalized metric we shall mean a pseudo-Riemannian metric tensor in our pull-back bundle. The main new results are contained in Section 5. We shall say, informally, that a metric covariant derivative is 'good' if it is related in a natural way to an Ehresmann connection determined by the metric alone. We shall find a family of good metric derivatives for the so-called weakly normal Moór-Vanstone metrics and a distinguished good metric derivative for a certain class of Miron metrics.
We study metrics on the pullback bundle of a tangent bundle by its own projection. We investigate... more We study metrics on the pullback bundle of a tangent bundle by its own projection. We investigate the circumstances under which an arbitrary metric admits a regular Lagrangian and thus an associated semispray. We present a simple coordinate-free formulation for all metric derivatives.
We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forwar... more We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forward) complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces. More precisely, we show that under these hypotheses the Finsler manifold is isometric to the tangent Minkowski vector space of the fixed point of the homothety via the exponential map of the canonical spray of the Finsler manifold.
A large class of special Finsler manifolds can be endowed with Finsler connections whose \h-part"... more A large class of special Finsler manifolds can be endowed with Finsler connections whose \h-part" does not depend on the directions. We call these Finsler connections h-basic and present a systematic treatment of them, using (in a simpli¯ed form) the FrÄ olicher-Nijenhuis calculus. We provide an axiomatic description of a distinguished class of h-basic Finsler connections, the class of Ichijy¹ o connections. With the help of an Ichijy¹ o connection we present new characterizations of generalized Berwald manifolds, as well as { in particular { of Berwald manifolds and locally Minkowski manifolds.
After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie de... more After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-free covariant derivative on the base manifold compatible with the Finsler function and Aikou's characterization of Berwald manifolds. Finally, we study some implications of V. Matveev's observation according to which quadratic convexity may be omitted from the definition of a Berwald manifold. These include, among others, a generalization of Z. I. Szabó's wellknown metrization theorem, and leads also to a natural generalization of Berwald manifolds, to Berwald-Matveev manifolds.
Introduction 1 Background 2 Calculus in topological vector spaces and beyond 3 The Chern-Rund der... more Introduction 1 Background 2 Calculus in topological vector spaces and beyond 3 The Chern-Rund derivative
Differential Geometry and Its Applications - Proceedings of the 10th International Conference on DGA2007, 2008
We sketch an economical framework and a simple index-free cal- culus for direction-dependent obje... more We sketch an economical framework and a simple index-free cal- culus for direction-dependent objects, based almost exclusively on Berwald derivative. To illustrate the efficiency of these tools, we characterize projec- tively Finslerian sprays, and derive Hamel's PDEs in an analytic version of Hilbert's fourth problem.
Curvature collineations of a spray manifold induced by the Lie symmetries of the underlying spray... more Curvature collineations of a spray manifold induced by the Lie symmetries of the underlying spray are studied. The basic observation is that the Jacobi endomorphism and the Berwald curvature are invariant under these symmetries; this implies the invariance of the further curvature data. Our main technical tool is an appropriate Lie derivative operator along the tangent bundle projection.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz HORIZONTAL MAPS WITH HOMOGENEITY CONDITION József Szilasi XX X X p-linear map. If specially П € A^^B;^), then i(x)ň Є SecÇ, namely (І(X)Я)(X):= ß x [x(x)]. 3. Vertical subbundle. The differential of the projection ir : E-* B is a strong bundle map dir : т E-*• 1.^ of constant rank, hence
Generalized Berwald spaces are the Finsler spaces which admit metric linear connections in the ta... more Generalized Berwald spaces are the Finsler spaces which admit metric linear connections in the tangent bundle of their base manifold. They reduce to Berwald spaces if the torsion of this connection vanishes. We show that generalized Berwald spaces with base manifold R n are exactly the affine deformations of Minkowski spaces. They can also be represented as a pair of a Riemannian and a Minkowski space, and they coincide with the Finsler spaces of 1-form metric. We investigate their reduction to Minkowski, Riemannian or Euclidean spaces, as well as their conformal relations.
In this paper, we give an equivalent characterization of conformal vector fields on a Finsler man... more In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold [Formula: see text], whose metric [Formula: see text] is defined by a Riemannian metric [Formula: see text] and a 1-form [Formula: see text]. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds [Formula: see text] when [Formula: see text] is of constant sectional curvature and [Formula: see text] is a conformal 1-form with respect to [Formula: see text].
After a careful study of the mixed curvatures of the Berwald-type (in particular, Berwald) connec... more After a careful study of the mixed curvatures of the Berwald-type (in particular, Berwald) connections, we present an axiomatic description of the so-called Yano-type Finsler connections. Using the Yano connection, we derive an intrinsic expression of Douglas' famous projective curvature tensor and we also represent it in terms of the Berwald connection. Utilizing a clever observation of Z. Shen, we show in a coordinate-free manner that a "spray manifold" is projectively equivalent to an affinely connected manifold iff its Douglas tensor vanishes. From this result we infer immediately that the vanishing of the Douglas tensor implies that the projective Weyl tensor of the Berwald connection "depends only on the position".
We prove that a dieomorphism of a manifold with an Ehresmann connection is an automorphism of the... more We prove that a dieomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesics, considered as parametrized curves, to geodesics) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative).
In this survey we approach some aspects of tangent bundle geometry from a new viewpoint. After an... more In this survey we approach some aspects of tangent bundle geometry from a new viewpoint. After an outline of our main tools, i.e., the pull-back bundle formalism, we give an overview of Ehresmann connections and covariant derivatives in the pull-back bundle of a tangent bundle over itself. Then we define and characterize some special classes of generalized metrics. By a generalized metric we shall mean a pseudo-Riemannian metric tensor in our pull-back bundle. The main new results are contained in Section 5. We shall say, informally, that a metric covariant derivative is 'good' if it is related in a natural way to an Ehresmann connection determined by the metric alone. We shall find a family of good metric derivatives for the so-called weakly normal Moór-Vanstone metrics and a distinguished good metric derivative for a certain class of Miron metrics.
We study metrics on the pullback bundle of a tangent bundle by its own projection. We investigate... more We study metrics on the pullback bundle of a tangent bundle by its own projection. We investigate the circumstances under which an arbitrary metric admits a regular Lagrangian and thus an associated semispray. We present a simple coordinate-free formulation for all metric derivatives.
We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forwar... more We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forward) complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces. More precisely, we show that under these hypotheses the Finsler manifold is isometric to the tangent Minkowski vector space of the fixed point of the homothety via the exponential map of the canonical spray of the Finsler manifold.
A large class of special Finsler manifolds can be endowed with Finsler connections whose \h-part"... more A large class of special Finsler manifolds can be endowed with Finsler connections whose \h-part" does not depend on the directions. We call these Finsler connections h-basic and present a systematic treatment of them, using (in a simpli¯ed form) the FrÄ olicher-Nijenhuis calculus. We provide an axiomatic description of a distinguished class of h-basic Finsler connections, the class of Ichijy¹ o connections. With the help of an Ichijy¹ o connection we present new characterizations of generalized Berwald manifolds, as well as { in particular { of Berwald manifolds and locally Minkowski manifolds.
After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie de... more After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-free covariant derivative on the base manifold compatible with the Finsler function and Aikou's characterization of Berwald manifolds. Finally, we study some implications of V. Matveev's observation according to which quadratic convexity may be omitted from the definition of a Berwald manifold. These include, among others, a generalization of Z. I. Szabó's wellknown metrization theorem, and leads also to a natural generalization of Berwald manifolds, to Berwald-Matveev manifolds.
Introduction 1 Background 2 Calculus in topological vector spaces and beyond 3 The Chern-Rund der... more Introduction 1 Background 2 Calculus in topological vector spaces and beyond 3 The Chern-Rund derivative
Differential Geometry and Its Applications - Proceedings of the 10th International Conference on DGA2007, 2008
We sketch an economical framework and a simple index-free cal- culus for direction-dependent obje... more We sketch an economical framework and a simple index-free cal- culus for direction-dependent objects, based almost exclusively on Berwald derivative. To illustrate the efficiency of these tools, we characterize projec- tively Finslerian sprays, and derive Hamel's PDEs in an analytic version of Hilbert's fourth problem.
Curvature collineations of a spray manifold induced by the Lie symmetries of the underlying spray... more Curvature collineations of a spray manifold induced by the Lie symmetries of the underlying spray are studied. The basic observation is that the Jacobi endomorphism and the Berwald curvature are invariant under these symmetries; this implies the invariance of the further curvature data. Our main technical tool is an appropriate Lie derivative operator along the tangent bundle projection.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz HORIZONTAL MAPS WITH HOMOGENEITY CONDITION József Szilasi XX X X p-linear map. If specially П € A^^B;^), then i(x)ň Є SecÇ, namely (І(X)Я)(X):= ß x [x(x)]. 3. Vertical subbundle. The differential of the projection ir : E-* B is a strong bundle map dir : т E-*• 1.^ of constant rank, hence
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