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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

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Matematika- és Számítástudományok Doktori Iskola Differenciálgeometria és alkalmazásai programja . 18 Summary. 91. 19 Magyar nyelv¶ H. Akbar-Zadeh and J. Grifone works ([2], [3], [12], [13]) are worth mentioning. Grifone

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