ʹ͍ͭͯɺۂઢ্ͷ֤ʹ͓ ͚ΔҠಈํͷϕΫτϧ ˙ p(t) := ˙ xi(t) ∂ ∂xi Λߟ͑Δɻ͜͜Ͱɺxi(t) ۂઢ C Λہॴ࠲ඪͰදࣔͨ͠ ͷͰɺ ˙ xi(t) ύϥϝʔλ t ʹΑΔඍΛදΘ͢ɻ ͜ͷۂઢ্Ͱఆٛ͞ΕͨϕΫτϧ Z ͕ ∇ ˙ p(t) Z = 0 (8) Λຬͨ࣌͢ɺϕΫτϧ Z C ʹͦͬͯฏߦͰ͋Δͱݴ͏ɻ (8) Λہॴ࠲ඪදࣔ͢ΔͱɺܭࢉʹΑΓɺ͕࣍ࣜಘΒΕ·͢ɻ ˙ Zk(t) + Γ k ij (p(t)) ˙ xi(t)Zj(t) = 0 (9) ͜͜ͰɺZk(t) ɺۂઢ C ʹͦͬͨϕΫτϧͷΛ t Λύϥϝʔλͱͯ͠දࣔͨ͠ͷͰɺ ˙ Zk(t) ɺ ύϥϝʔλ t ʹΑΔඍΛද͠·͢ɻ ͦͯ͠ɺ2 ͭͷϕΫτϧΛಉ͡ۂઢʹͦͬͯฏߦҠಈͤͨ࣌͞ʹɺ͜ΕΒͷੵͷ͕มԽ͠ͳ͍ͱ͍͏ ݅Λଓʹ՝͢͜ͱ͕Ͱ͖·͢ɻ͜ͷΑ͏ͳ݅Λຬͨ͢ଓɺܭྔతͰ͋Δͱݴ͍·͢ɻ͜ͷ݅ɺ ଓΛఆٛ͢Δࡍʹඞਢͱ͍͏Θ͚Ͱ͋Γ·ͤΜ͕ɺRiemann ଟ༷ମͰɺલఏ݅ͷ 1 ͭͱͯ͠ɺଓ ͕ܭྔతͰ͋Δ͜ͱ͕՝ͤΒΕ·͢ɻۂઢ p(t) ʹͦͬͯฏߦͳϕΫτϧ Y ͱ Z Λߟ͑Δͱɺ͜ͷ݅ ࣍ࣜͰද͞Ε·͢ɻ d dt { gij (p(t))Y i(t)Zj(t) } = 0 ࠨลͷඍΛల։ͯ͠ (9) Λ༻͍Δͱɺ͜Ε࣍ࣜͱಉʹͳΓ·͢ɻ ( ∂gij ∂xk − Γki,j − Γkj,i ) ˙ xkY iZj = 0 ͜Ε͕ҙͷ xk, Y i, Zj ͰΓཱͭ͜ͱ͔Βɺଓ͕ܭྔతͰ͋Δඞཁे݅ɺ࣍ࣜͰ༩͑ΒΕΔ͜ ͱʹͳΓ·͢ɻ ∂gij ∂xk = Γki,j + Γkj,i (10) ͜Εɺ࣍ͷؔࣜΛہॴ࠲ඪܥͰදࣔͨ͠ͷʹҰக͢Δ͜ͱ͕͔Γ·͢ɻ ∀X, Y, Z ∈ X(M); Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇X Z) (11) 2.8 ۂςϯιϧͱᎇςϯιϧ ϢʔΫϦουۭؒͰɺดۂઢʹͦͬͯϕΫτϧΛฏߦҠಈ͢Δͱɺग़ൃʹͬͯདྷͨϕΫτϧɺ ࠷ॳͷϕΫτϧʹҰக͠·͢ɻҰํɺҰൠͷଓΛ࣋ͬͨଟ༷ମͰɺඞͣͦ͠ͷΑ͏ʹͳΓ·ͤΜɻ Riemann زԿֶΛ͔ͬͨ͡ࣄͷ͋Δํɺٿͷද໘্ͰϕΫτϧΛฏߦҠಈ͢Δྫ͕͙͢ʹ಄ʹࢥ͍ු͔ Ϳ͜ͱͰ͠ΐ͏ɻ ͜ΕΛඍখͳฏߦ࢛ลܗ্ͷҠಈʹ͍ͭͯදݱͨ͠ͷ͕ۂςϯιϧͰ͢ɻ͋Δ p ʹ͓͍ͯɺϕΫτ ϧ X ͱ Y ͷํʹ͔ͬͨہॴ࠲ඪܥΛ༻ҙͯ͠ɺp = (x, y), (x+|X|, y), (x, y +|Y |), (x+|X|, y +|Y |) ͷ 4 Λߟ͑·͢ɻ͜͜Ͱɺ࠷ॳͷ 2 ͭͷ࠲ඪ͕ X ͱ Y ͷํʹରԠ͓ͯ͠Γɺ3 ͭΊҎ߱ͷ࠲ඪఆ 7