1. notice
  3. ไธญๆ–‡
  4. 1. feature
  6. ้€ป่พ‘
  7. 2. ้€ป่พ‘
  8. 3. ้›†ๅˆ่ฎบ
  9. 4. ๆ˜ ๅฐ„
  10. 5. ๅบ
  11. 6. ็ป„ๅˆ
  13. ๅพฎ็งฏๅˆ†
  14. 7. ๅฎžๆ•ฐ
  15. 8. ๆ•ฐๅˆ—ๆž้™
  16. 9. โ„^n
  17. 10. Euclidean ็ฉบ้—ด
  18. 11. Minkowski ็ฉบ้—ด
  19. 12. ๅคš้กนๅผ
  20. 13. ่งฃๆž (Euclidean)
  21. 14. ่งฃๆž (Minkowski)
  22. 15. ่งฃๆž struct ็š„ๆ“ไฝœ
  23. 16. ๅธธๅพฎๅˆ†ๆ–น็จ‹
  24. 17. ไฝ“็งฏ
  25. 18. ็งฏๅˆ†
  26. 19. ๆ•ฃๅบฆ
  27. 20. ็ฝ‘ๆž้™
  28. 21. ็ดง่‡ด
  29. 22. ่ฟž้€š
  30. 23. ๆ‹“ๆ‰‘ struct ็š„ๆ“ไฝœ
  31. 24. ๆŒ‡ๆ•ฐๅ‡ฝๆ•ฐ
  32. 25. ่ง’ๅบฆ
  34. ๅ‡ ไฝ•
  35. 26. ๆตๅฝข
  36. 27. ๅบฆ่ง„
  37. 28. ๅบฆ่ง„็š„่”็ปœ
  38. 29. Levi-Civita ๅฏผๆ•ฐ
  39. 30. ๅบฆ่ง„็š„ๆ›ฒ็Ž‡
  40. 31. Einstein ๅบฆ่ง„
  41. 32. ๅธธๆˆช้ขๆ›ฒ็Ž‡
  42. 33. simple-symmetric-space
  43. 34. ไธปไธ›
  44. 35. ็พคไฝœ็”จ
  45. 36. ็ƒๆžๆŠ•ๅฝฑ
  46. 37. Hopf ไธ›
  48. ๅœบ่ฎบ
  49. 38. ้ž็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  50. 39. ็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  51. 40. ็บฏ้‡ๅœบ
  52. 41. ็บฏ้‡ๅœบ็š„ๅฎˆๆ’ๆต
  53. 42. ้ž็›ธๅฏน่ฎบ็บฏ้‡ๅœบ
  54. 43. ๅ…‰้”ฅๅฐ„ๅฝฑ
  55. 44. ๆ—ถ็ฉบๅŠจ้‡็š„่‡ชๆ—‹่กจ็คบ
  56. 45. Lorentz ็พค
  57. 46. ๆ—‹้‡ๅœบ
  58. 47. ๆ—‹้‡ๅœบ็š„ๅฎˆๆ’ๆต
  59. 48. ็”ต็ฃๅœบ
  60. 49. ๅผ ้‡ๅœบ็š„ Laplacian
  61. 50. Einstein ๅบฆ่ง„
  62. 51. ็›ธไบ’ไฝœ็”จ
  63. 52. ่ฐๆŒฏๅญ้‡ๅญๅŒ–
  64. 53. ๆ—‹้‡ๅœบๆ‚้กน
  65. 54. ๅ‚่€ƒ
  67. English
  68. 55. notice
  69. 56. feature
  71. logic-topic
  72. 57. logic
  73. 58. set-theory
  74. 59. map
  75. 60. order
  76. 61. combinatorics
  78. calculus
  79. 62. real-numbers
  80. 63. limit-sequence
  81. 64. โ„^n
  82. 65. Euclidean-space
  83. 66. Minkowski-space
  84. 67. polynomial
  85. 68. analytic-Euclidean
  86. 69. analytic-Minkowski
  87. 70. analytic-struct-operation
  88. 71. ordinary-differential-equation
  89. 72. volume
  90. 73. integral
  91. 74. divergence
  92. 75. limit-net
  93. 76. compact
  94. 77. connected
  95. 78. topology-struct-operation
  96. 79. exponential
  97. 80. angle
  99. geometry
  100. 81. manifold
  101. 82. metric
  102. 83. metric-connection
  103. 84. geodesic-derivative
  104. 85. curvature-of-metric
  105. 86. Einstein-metric
  106. 87. constant-sectional-curvature
  107. 88. simple-symmetric-space
  108. 89. principal-bundle
  109. 90. group-action
  110. 91. stereographic-projection
  111. 92. Hopf-bundle
  113. field-theory
  114. 93. point-particle-non-relativity
  115. 94. point-particle-relativity
  116. 95. scalar-field
  117. 96. scalar-field-current
  118. 97. scalar-field-non-relativity
  119. 98. projective-lightcone
  120. 99. spacetime-momentum-spinor-representation
  121. 100. Lorentz-group
  122. 101. spinor-field
  123. 102. spinor-field-current
  124. 103. electromagnetic-field
  125. 104. Laplacian-of-tensor-field
  126. 105. Einstein-metric
  127. 106. interaction
  128. 107. harmonic-oscillator-quantization
  129. 108. spinor-field-misc
  130. 109. reference

note-math

cf. wiki:Symmetric_space wiki:Simple_Lie_group

[symmetric-space-locally] :=

Example quadratic manifold, simple-Lie-group and related symmetric-space

constant-sectional-curvature ==> symmetric-space

simple-Lie-group := Lie-algebra & Lie-bracket cannot decompose

Killing form := for ๅค„็š„ๅˆ‡็ฉบ้—ด็š„

็„ถๅŽๅฎšไน‰ ๅค„็š„ metric ้€š่ฟ‡ action ็”Ÿๆˆ็š„ๅ…ถๅฎƒๅœฐๆ–น็š„ metric, ่€Œไธ”ๆ˜ฏ bi-invariant ็š„ i.e. ็พคไฝœ็”จ็š„ไธค็งๅฝขๅผ้ƒฝ็ป™ๅ‡บ็›ธๅŒ็š„ metric

่ฟ™ๆ ท็š„ๅฎšไน‰ไฝฟๅพ—็พคไฝœ็”จๆ˜ฏ Killing-form ็š„ isometry

ไธๆ˜ฏ simple-Lie-group ไนŸๅฏไปฅๅฎšไน‰ Killing-form

Question Killing-form ็š„ๅฎšไน‰็š„ๅŠจๆœบ?

simple-Lie-group <==> Killing-form ้ž้€€ๅŒ–

simple-Lie-group and its symmetric-space ็š„ Killing-form ๆ˜ฏ Einstein-metric

Proof of simple-Lie-group ็š„ๆƒ…ๅ†ต

  • for Lie algebra

Proof

Lie-algebra ==> ฮด-isometry ==> for ฮด-group-action ,

ๅ› ไธบ Killing-form ๆ˜ฏ็พคไฝœ็”จ็”Ÿๆˆ็š„ metric ๆ‰€ไปฅ Lie-derivative ๆ˜ฏ้›ถ

ๅฏนไบŽ ็”Ÿๆˆ็š„ๅœบ

  • geodesic-derivative . Proof see below

  • curvature

  • . hence symmetric-space-locally

  • curvature

  • sectional-curvature for orthonormal

  • Ricci-curvature . hence Einstein-metric

  • scalar-curvature

Prop at , ๅŒ็†ๅฏน ็”Ÿๆˆ็š„ๅœบ (bi-invariant)

Proof

Prop

่ฟ™็ป™ๅ‡บ

with , ่ฟ™็ป™ๅ‡บ

Proof of

need

็”ฑไบŽ็พคไฝœ็”จ็”Ÿๆˆ , ๅธธๅ€ผ ==>

need

need

by

Question any more intuitive proof?

Prop for simple-Lie-group

Lie algebra ็”Ÿๆˆ็š„ bi-invariant vector field ็š„็งฏๅˆ†ๆ›ฒ็บฟ้ƒฝๆ˜ฏ Killing-form ๆต‹ๅœฐ็บฟ, ๅ› ไธบ

  • ๆต‹ๅœฐ็บฟๅฏไปฅๅ†™ไธบ
  • ๅ‡่ฎพ ๆ˜ฏ ็š„็งฏๅˆ†ๆ›ฒ็บฟ

ไบŒๆฌกๅž‹ๆตๅฝข. ็š„ๅฏน็งฐ็พค

  • orbit type or

    • induced metric signature (normal vector )
    • isotropy-group
    • quotient
    • isometry of is (isometry ๅ‡่ฎพไฟๆŒๆ–นๅ‘)

  • orbit type or

    • induced metric signature (normal vector )
    • isotropy-group
    • quotient
    • isometry of is

Example

  • spatial manifold ๆœ‰

  • ๆ—ถ็ฉบไบŒๆฌกๅž‹ๆตๅฝขๆœ‰ ๅ’Œๅ•ๅถๅŒๆ›ฒ้ข

ไบŒๆฌกๅž‹ๆตๅฝข็š„ไพ‹ๅญๆœ‰่ฟ™็งๆ€ง่ดจ

simple-Lie-group , simple-Lie-group isotropy , orbit

Lie-algebra ๆœ‰ๆญฃไบคๅˆ†่งฃ , ไธๆ˜ฏ Lie bracket ๅˆ†่งฃ

ๆ˜ฏ ็š„ Lie-algebra, ๆ˜ฏๆญฃไบค่กฅ

็ป™ๅ‡บ ็š„ๅๆ ‡

็š„ Killing-form ๅฏผๅ‡บ ็š„ Killing-form ๅ’Œ ็š„ Einstein metric