lie derivatives and geometric vector fields in spray and finsler geometry

115 Pages·2015·0.74 MB·Hungarian

DE TTK
1949
LIE DERIVATIVES AND
GEOMETRIC VECTOR FIELDS
IN SPRAY AND FINSLER GEOMETRY
Egyetemi doktori (PhD) ØrtekezØs
T(cid:243)th Anna
TØmavezetfi: Dr. Szilasi J(cid:243)zsef
Debreceni Egyetem
TermØszettudomÆnyi Doktori TanÆcs
Matematika- Øs SzÆm(cid:237)tÆstudomÆnyok Doktori Iskola
Debrecen, 2015.

Ezen ØrtekezØst a Debreceni Egyetem TermØszettudomÆnyi Doktori TanÆcs
Matematika- Øs SzÆm(cid:237)tÆstudomÆnyok Doktori Iskola Di(cid:27)erenciÆlgeometria
Øs alkalmazÆsai programja keretØben kØsz(cid:237)tettem a Debreceni Egyetem ter-
mØszettudomÆnyi doktori (PhD) fokozatÆnak elnyerØse cØljÆb(cid:243)l.
Debrecen, 2015 ..................
.....................
T(cid:243)th Anna
doktorjel(cid:246)lt
Tanœs(cid:237)tom, hogy T(cid:243)th Anna doktorjel(cid:246)lt 2008-2011 k(cid:246)z(cid:246)tt a fent meg-
nevezett Doktori Iskola Di(cid:27)erenciÆlgeometria Øs alkalmazÆsai programjÆ-
nak keretØben irÆny(cid:237)tÆsommal vØgezte munkÆjÆt. Az ØrtekezØsben foglalt
eredmØnyekhez a jel(cid:246)lt (cid:246)nÆll(cid:243) alkot(cid:243) tevØkenysØgØvel meghatÆroz(cid:243)an hoz-
zÆjÆrult. Az ØrtekezØs elfogadÆsÆt javasolom.
Debrecen, 2015 ..................
.....................
Dr. Szilasi J(cid:243)zsef
tØmavezetfi

Lie derivatives and geometric vector (cid:28)elds
in spray and Finsler geometry
(cid:201)rtekezØs a doktori (Ph.D.) fokozat megszerzØse ØrdekØben a
matematika tudomÆnyÆgban
˝rta: T(cid:243)th Anna okleveles matematika-kØmia szakos tanÆr
KØsz(cid:252)lt a Debreceni Egyetem
Matematika- Øs SzÆm(cid:237)tÆstudomÆnyok Doktori IskolÆja
Di(cid:27)erenciÆlgeometria Øs alkalmazÆsai programja keretØben
TØmavezetfi: Dr. Szilasi J(cid:243)zsef egyetemi docens
A doktori szigorlati bizottsÆg:
eln(cid:246)k: Dr..................... .....................
tagok: Dr..................... .....................
Dr..................... .....................
A doktori szigorlat idfipontja: 2013............
Az ØrtekezØs b(cid:237)rÆl(cid:243)i:
Dr. ..................... .....................
Dr. ..................... .....................
Dr. ..................... .....................
A b(cid:237)rÆl(cid:243)bizottsÆg:
eln(cid:246)k: Dr..................... .....................
tagok: Dr..................... .....................
Dr..................... .....................
Dr..................... .....................
Dr..................... .....................
Az ØrtekezØs vØdØsØnek idfipontja: 2015............

ACKNOWLEDGEMENT
I would like to express my sincere gratitude and thanks to my
supervisor, Dr. J(cid:243)zsef Szilasi for the continuous support of my studies
(from the beginning of my undergraduate studies to the end of my
Ph.D. studies). This Thesis appears in its current form due to his
motivation, patience, encouragement and relevant comments.
This Thesis could not have been written without the assistance and
guidance of my colleagues. I would therefore like to o(cid:27)er my sincere
thanks to all of them.
Special thanks to my family and my friends. I am grateful for all
thesacri(cid:28)cestheyhavemadetosupportme. Thankstoallmybrothers
and sisters in Christ for all the prayers said for me.
I praise God, the Almighty for providing me with this opportunity
and granting me the capability to proceed successfully.
Soli Deo Gloria!
‘I will go before you
and will level the mountains;
I will break down gates of bronze
and cut through bars of iron.’
/Isaiah 45:2/

Contents
1 Introduction 7
I Preliminary material 9
2 Manifolds and bundles 9
3 Tangent bundle and vector (cid:28)elds 13
4 Integral curves and (cid:29)ows 16
5 Tensor (cid:28)elds and di(cid:27)erential forms 20
6 Covariant derivatives 23
7 Constructions on the tangent bundle 26
II Lie derivatives in Finslerian setting 32
8 Finsler bundles 32
9 Vertical calculus 36
10 The classical Lie derivative 37
11 The Finslerian Lie derivative 42
III Lie symmetries 47
12 Semisprays and sprays 47
13 H-Killing vector (cid:28)elds 52
14 Curvature collineations in a spray manifold 65
IV Geometric vector (cid:28)elds on Finsler manifolds 73
15 Basic objects of a Finsler manifold 73
16 Killing vector (cid:28)elds on a Finsler manifold 78
17 Conformal and projective vector (cid:28)elds 84
V Summaries 91
18 Summary 91
19 Magyar nyelv¶ (cid:246)sszefoglal(cid:243) 100
References 109