Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications is a rich and self-contained exposition of recent developments in Riemannian submersions and maps relevant to complex geometry, focusing particularly on novel submersions, Hermitian manifolds, and K\{a}hlerian manifolds.
Riemannian submersions have long been an effective tool to obtain new manifolds and compare certain manifolds within differential geometry. For complex cases, only holomorphic submersions function appropriately, as discussed at length in Falcitelli, Ianus and Pastore’s classic 2004 book.
In this new book, Bayram Sahin extends the scope of complex cases with wholly new submersion types, including Anti-invariant submersions, Semi-invariant submersions, slant submersions, and Pointwise slant submersions, also extending their use in Riemannian maps.
The work obtains new properties of the domain and target manifolds and investigates the harmonicity and geodesicity conditions for such maps. It also relates these maps with discoveries in pseudo-harmonic maps. Results included in this volume should stimulate future research on Riemannian submersions and Riemannian maps.
1. Cover image - p. iv 2. Title page - p. iv 3. Table of Contents - p. iv 4. Copyright - p. iv 5. Dedication - p. v 6. Acknowledgments - p. ix 7. Preface - p. xi 8. About the Author - p. xvii 9. Chapter 1: Basic Geometric Structures on Manifolds - p. 1 10. Abstract - p. 1 11. 1 Riemannian manifolds and related topics - p. 1 12. 2 Vector bundles - p. 16 13. 3 Riemannian submanifolds and distributions - p. 19 14. 4 Riemannian submersions - p. 25 15. 5 Certain product structures on manifolds - p. 29 16. 6 Geometric structures along a map - p. 39 17. Chapter 2: Applications of Riemannian Submersions - p. 55 18. Chapter 3: Riemannian submersions From Almost Hermitian Manifolds - p. 73 19. Abstract - p. 73 20. 1 Almost Hermitian manifolds - p. 74 21. 2 Holomorphic submersions and invariant submersions - p. 83 22. 3 Anti-invariant Riemannian submersions - p. 85 23. 4 Semi-invariant submersions - p. 98 24. 5 Generic Riemannian submersions - p. 109 25. 6 Slant submersions - p. 118 26. 7 Semi-slant submersions - p. 126 27. 8 Hemi-slant submersions - p. 135 28. 9 Pointwise slant submersions - p. 147 29. 10 Einstein metrics on the total space of an anti-invariant submersion - p. 155 30. 11 Clairaut submersions from almost Hermitian manifolds - p. 159 31. Chapter 4: Riemannian Maps - p. 179 32. Abstract - p. 179 33. 1 Riemannian maps - p. 179 34. 2 Geometric structures along Riemannian maps - p. 185 35. 3 Totally geodesic Riemannian maps - p. 190 36. 4 Umbilical Riemannian maps - p. 197 37. 5 Harmonicity of Riemannian maps - p. 203 38. 6 Clairaut Riemannian maps - p. 207 39. 7 Circles along Riemannian maps - p. 210 40. 8 Chen first inequality for Riemannian maps - p. 214 41. 9 Einstein metrics on the total space of a Riemannian map - p. 219 42. Chapter 5: Riemannian Maps From Almost Hermitian Manifolds - p. 225 43. Abstract - p. 225 44. 1 Holomorphic Riemannian maps from almost Hermitian manifolds - p. 225 45. 2 Anti-invariant Riemannian maps from almost Hermitian manifolds - p. 229 46. 3 Semi-invariant Riemannian maps from almost Hermitian manifolds - p. 238 47. 4 Generic Riemannian maps from almost Hermitian manifolds - p. 247 48. 5 Slant Riemannian maps from almost Hermitian manifolds - p. 254 49. 6 Semi-slant Riemannian maps from almost Hermitian manifolds - p. 262 50. 7 Hemi-slant Riemannian maps from almost Hermitian manifolds - p. 271 51. Chapter 6: Riemannian Maps To Almost Hermitian Manifolds - p. 281 52. Abstract - p. 281 53. 1 Invariant Riemannian maps to almost Hermitian manifolds - p. 281 54. 2 Anti-invariant Riemannian maps to almost Hermitian manifolds - p. 287 55. 3 Semi-invariant Riemannian maps to almost Hermitian manifolds - p. 290 56. 4 Generic Riemannian maps to Kähler manifolds - p. 297 57. 5 Slant Riemannian maps to Kähler manifolds - p. 301 58. 6 Semi-slant Riemannian maps to Kähler manifolds - p. 308 59. 7 Hemi-slant Riemannian maps to Kähler manifolds - p. 318 60. Bibliography - p. 325 61. Index - p. 337
Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications is a rich and self-contained exposition of recent developments in Riemannian submersions and maps relevant to complex geometry, focusing particularly on novel submersions, Hermitian manifolds, and K\{a}hlerian manifolds.
Riemannian submersions have long been an effective tool to obtain new manifolds and compare certain manifolds within differential geometry. For complex cases, only holomorphic submersions function appropriately, as discussed at length in Falcitelli, Ianus and Pastore’s classic 2004 book.
In this new book, Bayram Sahin extends the scope of complex cases with wholly new submersion types, including Anti-invariant submersions, Semi-invariant submersions, slant submersions, and Pointwise slant submersions, also extending their use in Riemannian maps.
The work obtains new properties of the domain and target manifolds and investigates the harmonicity and geodesicity conditions for such maps. It also relates these maps with discoveries in pseudo-harmonic maps. Results included in this volume should stimulate future research on Riemannian submersions and Riemannian maps.
1. Cover image - p. iv 2. Title page - p. iv 3. Table of Contents - p. iv 4. Copyright - p. iv 5. Dedication - p. v 6. Acknowledgments - p. ix 7. Preface - p. xi 8. About the Author - p. xvii 9. Chapter 1: Basic Geometric Structures on Manifolds - p. 1 10. Abstract - p. 1 11. 1 Riemannian manifolds and related topics - p. 1 12. 2 Vector bundles - p. 16 13. 3 Riemannian submanifolds and distributions - p. 19 14. 4 Riemannian submersions - p. 25 15. 5 Certain product structures on manifolds - p. 29 16. 6 Geometric structures along a map - p. 39 17. Chapter 2: Applications of Riemannian Submersions - p. 55 18. Chapter 3: Riemannian submersions From Almost Hermitian Manifolds - p. 73 19. Abstract - p. 73 20. 1 Almost Hermitian manifolds - p. 74 21. 2 Holomorphic submersions and invariant submersions - p. 83 22. 3 Anti-invariant Riemannian submersions - p. 85 23. 4 Semi-invariant submersions - p. 98 24. 5 Generic Riemannian submersions - p. 109 25. 6 Slant submersions - p. 118 26. 7 Semi-slant submersions - p. 126 27. 8 Hemi-slant submersions - p. 135 28. 9 Pointwise slant submersions - p. 147 29. 10 Einstein metrics on the total space of an anti-invariant submersion - p. 155 30. 11 Clairaut submersions from almost Hermitian manifolds - p. 159 31. Chapter 4: Riemannian Maps - p. 179 32. Abstract - p. 179 33. 1 Riemannian maps - p. 179 34. 2 Geometric structures along Riemannian maps - p. 185 35. 3 Totally geodesic Riemannian maps - p. 190 36. 4 Umbilical Riemannian maps - p. 197 37. 5 Harmonicity of Riemannian maps - p. 203 38. 6 Clairaut Riemannian maps - p. 207 39. 7 Circles along Riemannian maps - p. 210 40. 8 Chen first inequality for Riemannian maps - p. 214 41. 9 Einstein metrics on the total space of a Riemannian map - p. 219 42. Chapter 5: Riemannian Maps From Almost Hermitian Manifolds - p. 225 43. Abstract - p. 225 44. 1 Holomorphic Riemannian maps from almost Hermitian manifolds - p. 225 45. 2 Anti-invariant Riemannian maps from almost Hermitian manifolds - p. 229 46. 3 Semi-invariant Riemannian maps from almost Hermitian manifolds - p. 238 47. 4 Generic Riemannian maps from almost Hermitian manifolds - p. 247 48. 5 Slant Riemannian maps from almost Hermitian manifolds - p. 254 49. 6 Semi-slant Riemannian maps from almost Hermitian manifolds - p. 262 50. 7 Hemi-slant Riemannian maps from almost Hermitian manifolds - p. 271 51. Chapter 6: Riemannian Maps To Almost Hermitian Manifolds - p. 281 52. Abstract - p. 281 53. 1 Invariant Riemannian maps to almost Hermitian manifolds - p. 281 54. 2 Anti-invariant Riemannian maps to almost Hermitian manifolds - p. 287 55. 3 Semi-invariant Riemannian maps to almost Hermitian manifolds - p. 290 56. 4 Generic Riemannian maps to Kähler manifolds - p. 297 57. 5 Slant Riemannian maps to Kähler manifolds - p. 301 58. 6 Semi-slant Riemannian maps to Kähler manifolds - p. 308 59. 7 Hemi-slant Riemannian maps to Kähler manifolds - p. 318 60. Bibliography - p. 325 61. Index - p. 337
