Connections, sprays, and Finsler structures:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2014
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XXI, 709 S. |
| ISBN: | 9789814440097 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV041153756 | ||
| 003 | DE-604 | ||
| 005 | 20220527 | ||
| 007 | t | ||
| 008 | 130718s2014 |||| 00||| eng d | ||
| 020 | |a 9789814440097 |9 978-981-4440-09-7 | ||
| 035 | |a (OCoLC)814455493 | ||
| 035 | |a (DE-599)HBZHT017699455 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-703 | ||
| 082 | 0 | |a 516.36 |2 23 | |
| 084 | |a SK 370 |0 (DE-625)143234: |2 rvk | ||
| 100 | 1 | |a Szilasi, József |e Verfasser |0 (DE-588)1043382739 |4 aut | |
| 245 | 1 | 0 | |a Connections, sprays, and Finsler structures |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2014 | |
| 300 | |a XXI, 709 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Finsler-Geometrie |0 (DE-588)4451048-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Finsler-Geometrie |0 (DE-588)4451048-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Lovas, Rezső L. |e Verfasser |0 (DE-588)1043382879 |4 aut | |
| 700 | 1 | |a Kertész, Dávid Cs. |e Verfasser |0 (DE-588)1043382992 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-026129169 | ||
Datensatz im Suchindex
| _version_ | 1804150557043064832 |
|---|---|
| adam_text | Contents
Preface
Acknowledgments
xiii
1.
Modules, Algebras and Derivations
1
1.1
Modules and Vector Spaces
................. 1
1.1.1
Basic Definitions and Facts
............. 1
1.1.2
Homomorphisms
................... 8
1.1.3
Cosets and
Affine
Mappings
............ 18
1.2
Tensors
............................ 20
1.2.1
Tensors as Multilinear Mappings
.......... 21
1.2.2
Substitution Operators and Pull-back
....... 22
1.2.3
Canonical Isomorphisms
.............. 23
1.2.4
Tensor Components
................. 24
1.2.5
Contraction and Trace
............... 27
1.3
Algebras and Derivations
.................. 30
1.3.1
Basic Definitions
................... 30
1.3.2
Derivations
....................., 31
1.3.3
Lie Algebras
..................... 32
1.3.4
Graded Algebras and Graded Derivations
..... 33
1.3.5
The Exterior Algebra of an R-module
....... 35
1.3.6
Determinants
.................... 39
1.3.7
Volume Forms and Orientation
........... 43
1.4
Orthogonal Spaces
...................... 44
1.4.1
Scalar Product and Non-degeneracy
........ 44
1.4.2
The Associated Quadratic Form
.......... 45
1.4.3
Orthonormal
Bases
................. 47
XV
xvi
Connections, Sprays and Finsler Structures
1.4.4
Orthogonal
Mappings and the
Adjoint
...... 49
1.4.5
Modules
with Scalar Product
............ 51
2.
Manifolds and Bundles
55
2.1
Smooth Manifolds and Mappings
.............. 55
2.1.1
Charts, Atlases, Manifolds
............. 55
2.1.2
Examples of Manifolds
............... 57
2.1-3
Mappings of Class Cr
................ 61
2.1.4
Smooth Partitions of Unity
............. 67
2.2
Fibre Bundles
......................... 68
2.2.1
Fibre Bundles, Bundle Maps, Sections
....... 68
2.2.2
Vector Bundles
.................... 74
2.2.3
Examples and Constructions
............ 82
2.2.4
π
-tensors
and
π
-tensor
fields
............ 89
2.2.5
Vector Bundles with Additional Structures
.... 91
3.
Vector Fields, Tensors and Integration
97
3.1
Tangent Bundle and Vector Fields
............. 97
3.1.1
Tangent Vectors and Tangent Space
........ 97
3.1.2
The Derivative of a Differentiable Mapping
.... 101
3.1.3
Some Local Properties of Differentiable Mappings
105
3.1.4
The Tangent Bundle of a Manifold
........ 106
3.1.5
The Lie Algebra of Vector Fields
.......... 116
3.2
First-order Differential Equations
.............. 122
3.2.1
Basic Existence and Uniqueness Theorems
.... 122
3.2.2
Integral Curves
................... 127
3.2.3
Flows
......................... 131
3.2.4
Commuting Flows
.................. 136
3.3
Tensors and Differential Forms
............... 141
3.3.1
The Cotangent Bundle of a Manifold
....... 141
3.3.2
Tensors on a Manifold
................ 144
3.3.3
Tensor Derivations
.................. 147
3.3.4
Differential Forms
.................. 151
3.3.5
The Classical Graded Derivations of A(M)
.... 155
3.3.6
The
FröHcher-Nijenhuis
Theorem
......... 160
3.4
Integration on Manifolds
................... 167
3.4.1
Orientable
Manifolds
................ 167
3.4.2
Integration of Top Forms
.............. 168
Contents xvii
3.4.3
Stokes
Theorem................... 174
4.
Structures on Tangent Bundles
179
4.1
Vector Bundles on
Τ Μ ...................
179
4.1.1
Finsler Bundles and Finsler Tensor Fields
..... 179
4.1.2
The Vector Bundle Structure of r+: TTM
->
Τ Μ
188
4.1.3
The Vertical Subbundle of TTM
.......... 195
4.1.4
Acceleration and Reparametrizations
....... 209
4.1.5
The Complete Lift of a Vector Field
........ 211
4.1.6
The Vertical Endomorphism of TTM
....... 220
4.1.7
Push-forwards
.................... 226
4.2
Homogeneity
......................... 228
4.2.1
Homogeneous Mappings of Vector Spaces
..... 228
4.2.2
Homogeneous Functions on TM
.......... 231
4.2.3
Homogeneous Vector Fields on
Τ Μ........
234
5.
Sprays and Lagrangians
237
5.1
Sprays and the Exponential Map
.............. 237
5.1.1
Second-order Vector Fields and Some of Their Mu¬
tants
......................... 237
5.1.2
Geodesies of
a Semispray
.............. 247
5.1.3
The Exponential Map
................ 253
5.1.4
The Theorem of Whitehead
............ 257
5.2 Lagrange
Functions
...................... 262
5.2.1
Regularity and Global Dynamics
.......... 262
5.2.2
First Variation
.................... 268
6.
Covariant Derivatives
277
6.1
Differentiation in Vector Bundles
.............. 277
6.1.1
Covariant Derivative on a Vector Bundle
..... 277
6.1.2
The Second Covariant Differential
......... 287
6.1.3
Exterior Covariant Derivative
........... 289
6.1.4
Metric Derivatives
.................. 291
6.1.5
Curvature and Torsion
............... 293
6.1.6
The Levi-Civita Derivative
............. 302
6.1.7
Covariant Derivative Along a Curve
........ 307
6.1.8
Parallel Translation with Respect to a Covariant
Derivative
......................
xviii
Connections, Sprays and Finsler Structures
6.1.9
Geodesies of an Affinely Connected Manifold .
. . 318
6.1.10
Hypersurfaces in a Riemannian
Manifold
..... 321
6.2
Covariant
Derivatives
on a
Finsłer
Bundle.........
326
6.2.1
Curvature and Torsion
............... 326
6.2.2
Deflection and Regularities
............. 330
6.2.3
Vertical Covariant Derivative Operators
...... 333
6.2.4
The Vertical Hessian of a Lagrangian
....... 340
6.2.5
Parallelism and Geodesies
............. 343
6.2.6
Metric f-covariant Derivatives
........... 347
7.
Theory of Ehresmann Connections
351
7.1
Horizontal Subbundles
.................... 351
7.2
Ehresmann Connections and Associated Objects
..... 357
7.3
Constructions of Ehresmann Connections
......... 368
7.3.1
Ehresmann Connections and Projection Operators
368
7.3.2
Ehresmann Connections from Regular Covariant
Derivatives
...................... 369
7.3.3
The Crampin-
Grifone
Construction
........ 371
7.4
Some Useful Technicalities
.................. 374
7.5
Homogeneity and Linearity
................. 378
7.5.1
Homogeneity Conditions
.............. 378
7.5.2
The Ehresmann Connection of an Affinely Con¬
nected Manifold
................... 383
7.5.3
The Linear Deviation
................ 387
7.6
Parallel Translation with Respect to an Ehresmann Con¬
nection
............................. 389
7.7
Geodesies of an Ehresmann Connection
.......... 395
7.8
Curvature and Torsion
.................... 396
7.9
Ehresmann Connections and Covariant Derivatives
.... 405
7.10
The Induced Berwald Derivative
.............. 411
7.11
The Debauch of Indices
................... 416
7.12
Tension, Torsion, Curvature and Geodesies Again
..... 421
7.13
The Berwald Curvature
................... 426
7.14
The
Affine
Curvature
..................... 431
7.15
Linear Ehresmann Connections Revisited
......... 439
Contents
XIX
8.
Geometry
of
Spray
Manifolds
445
8.1
The Berwald Connection and Related Constructions
. . . 445
8.1.1
The Berwald Connection
.............. 445
8.1.2
The Induced Berwald Derivative
.......... 446
8.1.3
Torsion and Curvature
............... 447
8.1.4
Coordinate Description
............... 448
8.2 Affine
Deviation
....................... 451
8.2.1
The Jacobi Endomorphism
............. 451
8.2.2
Jacobi Fields
..................... 460
8.3
The Weyl Endomorphism
.................. 463
8.4
Projective
Changes
...................... 472
8.4.1
Projectively Related Sprays
............ 472
8.4.2
Changes of Associated Objects
........... 475
8.4.3
Projectively Related Covariant Derivatives
.... 480
8.4.4
The Meaning of the Weyl Endomorphism
..... 482
8.4.5
The Douglas Tensor
................. 483
8.4.6
The Meaning of the Douglas Tensor
........ 487
8.5
Integrability and Flatness
.................. 494
9.
Finsler Norms and Finsler Functions
503
9.1
Finsler Vector Spaces
..................... 503
9.1.1
Convexity
...................... 503
9.1.2
Pre-Finsler Norms
.................. 512
9.1.3
Finsler Norms and Some of Their Characterizations
517
9.1.4
Reduction to Euclidean Vector Space
....... 524
9.1.5
Averaged Scalar Product on a Gauge Vector Space
527
9.2
Fundamentals on Finsler Functions
............. 532
9.2.1
Pre-Finsler Manifolds
................ 532
9.2.2
Finsler Functions and the Canonical Spray
.... 539
9.2.3
The
Rapcsák
Equations
............... 543
9.2.4
Riemannian Finsler Functions
........... 547
9.3
Notable Covariant Derivatives on a Finsler Manifold
. . . 552
9.3.1
The Fundamental Lemma of Finsler Geometry
. . 552
9.3.2
The Finslerian Berwald Derivative
......... 555
9.3.3
The Cartan Derivative
............... 563
9.3.4
The Chern-Rund and the Hashiguchi Derivative
564
9.4 Isotropie
Finsler Manifolds
................. 568
9.4.1
Characterizations of Isotropy
............ 568
xx
Connections, Sprays and Finsler Structures
9.4.2
The Flag Curvature
................. 573
9.4.3
The Generalized
Schur
Theorem
.......... 575
9.5
Geodesies and Distance
................... 580
9.5.1
Finslerian Geodesies and Isometries
........ 580
9.5.2
The Finslerian Distance
............... 585
9.5.3
The Myers
-
Steenrod Theorem
........... 589
9.6
Projective Relatedness
Again
................ 593
9.7
Projective Metrizability
................... 597
9.8
Berwald Manifolds
...................... 601
9.9
Oriented Finsler Surfaces
.................. 608
9.9.1
Berwald Frames
................... 608
9.9.2
The Fundamental Equations of Finsler Surfaces
. 610
9.9.3
Surviving Curvature Components
......... 616
9.9.4
Szabo s Rigidity Theorem
............. 619
Appendix A Sets, Mappings and Operations
625
A.I Set Notations and Concepts
................. 625
A.
2
Mappings
........................... 627
A.3 Groups and Group Actions
................. 632
A.4 Rings
............................. 640
Appendix
В
Topological Concepts
643
B.I Basic Definitions and Constructions
............ 643
B.2 Metric Topologies and the Contraction Principle
..... 645
B-3 More Topological Concepts
................. 647
B.4 Topological Vector Spaces
.................. 650
Appendix
С
Calculus in Vector Spaces
653
C.I Differentiation in Vector Spaces
............... 653
C.2 Canonical Constructions
................... 665
C.2.1 Tangent Bundle and Derivative
.......... 665
C.2.2 Lifts of Functions
.................. 669
C.2.3 The Vector Bundle
τ*:
TTU
-> TU........ 671
C.2.4 Lifts of Vector Fields
................ 673
C.2.5
б,
i, j
and
J
..................... 674
С.З
The Standard Covariant Derivative
............. 676
Contents
xxi
Bibliography
681
General Convent tans
687
Notation Index
689
índex
695
CONNECTIONS, SPRAYS AND
FINSLER STRUCTURES
This book provides a comprehensive
introduction
to
Finsier
geometry in the language of present-day mathematics. Through
Finsler geometry, it also introduces the reader to other
structures and techniques of differential geometry.
Prerequisites for reading the book are minimal: undergraduate
linear algebra (over the reals) and analysis. The necessary
concepts and tools of advanced linear algebra (over modules),
point set topology,
multivariable
calculus and the rudiments
of the theory of differential equations are integrated in the
text. Basic manifold and bundle theories are treated concisely,
carefully and (apart from proofs) in a self-contained manner.
The backbone of the book is the detailed and original exposition
of tangent bundle geometry,
Ehresmann
connections and
sprays. It turns out that these structures are important not only
in their own right and in the foundation of Finsler geometry,
but they can be also regarded as the cornerstones of the huge
edifice of Differential Geometry.
The authors emphasize the conceptual aspects, but carefully
elaborate calculative aspects as well (tensor derivations,
graded derivations and covariant derivatives). Although they
give preference to index-free methods, they also apply the
techniques of traditional tensor calculus.
Most proofs are elaborated in detail, which makes the book
suitable for self-study. Nevertheless, the authors provide
for more advanced readers as well by supplying them with
adequate material, and the book may also serve as a reference.
|
| any_adam_object | 1 |
| author | Szilasi, József Lovas, Rezső L. Kertész, Dávid Cs |
| author_GND | (DE-588)1043382739 (DE-588)1043382879 (DE-588)1043382992 |
| author_facet | Szilasi, József Lovas, Rezső L. Kertész, Dávid Cs |
| author_role | aut aut aut |
| author_sort | Szilasi, József |
| author_variant | j s js r l l rl rll d c k dc dck |
| building | Verbundindex |
| bvnumber | BV041153756 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)814455493 (DE-599)HBZHT017699455 |
| dewey-full | 516.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.36 |
| dewey-search | 516.36 |
| dewey-sort | 3516.36 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01641nam a2200361 c 4500</leader><controlfield tag="001">BV041153756</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20220527 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">130718s2014 |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9789814440097</subfield><subfield code="9">978-981-4440-09-7</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)814455493</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)HBZHT017699455</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-703</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.36</subfield><subfield code="2">23</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 370</subfield><subfield code="0">(DE-625)143234:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Szilasi, József</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1043382739</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Connections, sprays, and Finsler structures</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">New Jersey [u.a.]</subfield><subfield code="b">World Scientific</subfield><subfield code="c">2014</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XXI, 709 S.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Finsler-Geometrie</subfield><subfield code="0">(DE-588)4451048-2</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Finsler-Geometrie</subfield><subfield code="0">(DE-588)4451048-2</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Lovas, Rezső L.</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1043382879</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Kertész, Dávid Cs.</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1043382992</subfield><subfield code="4">aut</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Klappentext</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-026129169</subfield></datafield></record></collection> |
| id | DE-604.BV041153756 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:40:49Z |
| institution | BVB |
| isbn | 9789814440097 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026129169 |
| oclc_num | 814455493 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XXI, 709 S. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Szilasi, József Verfasser (DE-588)1043382739 aut Connections, sprays, and Finsler structures New Jersey [u.a.] World Scientific 2014 XXI, 709 S. txt rdacontent n rdamedia nc rdacarrier Finsler-Geometrie (DE-588)4451048-2 gnd rswk-swf Finsler-Geometrie (DE-588)4451048-2 s DE-604 Lovas, Rezső L. Verfasser (DE-588)1043382879 aut Kertész, Dávid Cs. Verfasser (DE-588)1043382992 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Szilasi, József Lovas, Rezső L. Kertész, Dávid Cs Connections, sprays, and Finsler structures Finsler-Geometrie (DE-588)4451048-2 gnd |
| subject_GND | (DE-588)4451048-2 |
| title | Connections, sprays, and Finsler structures |
| title_auth | Connections, sprays, and Finsler structures |
| title_exact_search | Connections, sprays, and Finsler structures |
| title_full | Connections, sprays, and Finsler structures |
| title_fullStr | Connections, sprays, and Finsler structures |
| title_full_unstemmed | Connections, sprays, and Finsler structures |
| title_short | Connections, sprays, and Finsler structures |
| title_sort | connections sprays and finsler structures |
| topic | Finsler-Geometrie (DE-588)4451048-2 gnd |
| topic_facet | Finsler-Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129169&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT szilasijozsef connectionsspraysandfinslerstructures AT lovasrezsol connectionsspraysandfinslerstructures AT kerteszdavidcs connectionsspraysandfinslerstructures |