❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ❛♥❞ ▲❛♣❧❛❝✐❛♥s ❢♦r ❣❡♥❡r❛❧✐③❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ■❛❦♦✈♦s ❆♥❞r♦✉❧✐❞❛❦✐s✮ ❯❢❛ ❋❡❞❡r❛❧ ❘❡s❡❛r❝❤ ❈❡♥tr❡ ❘❆❙✱ ❯❢❛✱ ❘✉ss✐❛ ▼✐❝r♦❧♦❝❛❧ ❛♥❞ ●❧♦❜❛❧ ❆♥❛❧②s✐s✱ ■♥t❡r❛❝t✐♦♥s ✇✐t❤ ●❡♦♠❡tr② ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ▼❛r❝❤ ✹✕✽✱ ✷✵✶✾ ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶ ✴ ✸✹
●❡♥❡r❛❧✐③❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s ❚❤❡ ❣♦❛❧✿ ●✐✈❡♥ ❛ ❝♦♠♣❛❝t s♠♦♦t❤ ♠❛♥✐❢♦❧❞ M ❛♥❞ ❛ ✭✈❡❝t♦r✮ ❞✐str✐❜✉t✐♦♥ D ✱ t♦ ❞❡✜♥❡ ❛♥❞ st✉❞② ♥❛t✉r❛❧ ❣❡♦♠❡tr✐❝ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ❛ss♦❝✐❛t❡❞ ✇✐t❤ ( M , D ) ✳ ❚❤❡ st❛♥❞❛r❞ ♥♦t✐♦♥ ♦❢ ❛ ❞✐str✐❜✉t✐♦♥ ✭❛ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥ ♦♥ M ♦❢ ❝♦♥st❛♥t r❛♥❦ p ✮✿ m ∈ M �→ D m ⊂ T m M , dim D m = p ✳ D m ❞❡♣❡♥❞s s♠♦♦t❤❧② ♦♥ m ✭❧♦❝❛❧❧②✱ t❤❡r❡ ✐s ❛ s♠♦♦t❤ ❢r❛♠❡ ♦❢ D ✮✳ ✭♦r✱ ❜r✐❡✢②✱ D ✐s ❛ r❛♥❦ p s♠♦♦t❤ s✉❜❜✉♥❞❧❡ ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TM ✮✳ ❖✉t❧✐♥❡✿ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r r❛♥❦✲✈❛r②✐♥❣ ✭♦r ❣❡♥❡r❛❧✐③❡❞✮ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s D ✳ ❋✐rst ♦❢ ❛❧❧✱ ✇❡ ❞✐s❝✉ss ▲❛♣❧❛❝✐❛♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ D ✿ ❆ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ ( M , D ) ✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧ d D : C ∞ ( M ) → C ∞ ( M , D ∗ ) ✳ ❚❤❡ ❛ss♦❝✐❛t❡❞ ▲❛♣❧❛❝✐❛♥ ∆ D = d ∗ D d D : C ∞ ( M ) → C ∞ ( M ) ✳ ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷ ✴ ✸✹
●❡♥❡r❛❧✐③❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s Pr❡❧✐♠✐♥❛r✐❡s M ❛ ❝♦♠♣❛❝t s♠♦♦t❤ ♠❛♥✐❢♦❧❞✱ X ( M ) = C ∞ ( M , TM ) t❤❡ s♣❛❝❡ ♦❢ s♠♦♦t❤ ✈❡❝t♦r ✜❡❧❞s ♦♥ M ❀ X ( M ) ✐s ❛ ▲✐❡✲❘✐♥❡❤❛rt ❛❧❣❡❜r❛✿ ❛ C ∞ ( M ) ✲♠♦❞✉❧❡✿ a ∈ C ∞ ( M ) , X ∈ X ( M ) �→ a · X ∈ X ( M ) , ❛ ▲✐❡ ❛❧❣❡❜r❛ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲✐❡ ❜r❛❝❦❡t✿ � � � ∂ Y i ∂ X i ∂ [ X , Y ] = XY − YX = X j − Y j ∂ x j ∂ x j ∂ x i i j j ❢♦r X = � ∂ x i , Y = � i X i ∂ i Y i ∂ ∂ x i ∈ X ( M ) ✳ ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✸ ✴ ✸✹
●❡♥❡r❛❧✐③❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s ▲❡t D ❜❡ ❛ C ∞ ( M ) ✲s✉❜♠♦❞✉❧❡ ♦❢ X c ( M ) ✿ ✶ ●✐✈❡♥ ❛♥ ♦♣❡♥ s✉❜s❡t U ♦❢ M ✱ ✇❡ ♣✉t ι U : U ֒ → M t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣✱ ❛♥❞✱ ❢♦r ❛ ✈❡❝t♦r ✜❡❧❞ X ∈ X ( M ) ✱ ✇r✐t❡ X | U = X ◦ ι U ✳ ❚❤❡ r❡str✐❝t✐♦♥ D | U ♦❢ D t♦ U ✐s t❤❡ C ∞ ( U ) ✲s✉❜♠♦❞✉❧❡ ♦❢ X c ( U ) ❣❡♥❡r❛t❡❞ ❜② f · X | U ✱ ✇❤❡r❡ f ∈ C ∞ c ( U ) ❛♥❞ X ∈ D ✳ ✷ ❲❡ s❛② t❤❛t t❤❡ ♠♦❞✉❧❡ D ✐s ❧♦❝❛❧❧② ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐❢✱ ❢♦r ❡✈❡r② x ∈ M ✱ t❤❡r❡ ❡①✐st ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ U ♦❢ x ❛♥❞ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ✈❡❝t♦r ✜❡❧❞s X 1 , . . . , X k ✐♥ X ( M ) s✉❝❤ t❤❛t D | U = C ∞ c ( U ) · X 1 | U + . . . + C ∞ c ( U ) · X k | U . ❲❡ s❛② t❤❛t t❤❡ ✈❡❝t♦r ✜❡❧❞s X 1 , . . . , X k ❣❡♥❡r❛t❡ D | U ♦r t❤❡② ❛r❡ ❧♦❝❛❧ ❣❡♥❡r❛t♦rs ♦❢ D ✳ ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✹ ✴ ✸✹
●❡♥❡r❛❧✐③❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s ●❡♥❡r❛❧✐③❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s ❉❡✜♥✐t✐♦♥ ❆ ✭❣❡♥❡r❛❧✐③❡❞✮ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥ ♦♥ M ✐s ❛ ❧♦❝❛❧❧② ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ C ∞ ( M ) ✲s✉❜♠♦❞✉❧❡ D ♦❢ t❤❡ C ∞ ( M ) ✲♠♦❞✉❧❡ X ( M ) ✳ ❈♦♥st❛♥t r❛♥❦ ❞✐str✐❜✉t✐♦♥s ●✐✈❡♥ ❛ r❛♥❦ p s♠♦♦t❤ s✉❜❜✉♥❞❧❡ D ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TM D ✐s t❤❡ s✉❜s♣❛❝❡ ♦❢ s♠♦♦t❤ ✈❡❝t♦r ✜❡❧❞s ♦♥ M ✱ t❛♥❣❡♥t t♦ D ✿ D = C ∞ ( M , D ) = { X ∈ X ( M ) : X ( m ) ∈ D m ∀ m ∈ M } . ■♥ t❤✐s ❝❛s❡✱ D ✐s ❛ ♣r♦❥❡❝t✐✈❡ C ∞ ( M ) ✲s✉❜♠♦❞✉❧❡ ♦❢ X ( M ) ✳ ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✺ ✴ ✸✹
❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s ❘❛♥❦ ✈❛r②✐♥❣ ❞✐str✐❜✉t✐♦♥s D ❛ ✭❣❡♥❡r❛❧✐③❡❞✮ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥ ♦♥ M ✭❛ ❧♦❝❛❧❧② ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ C ∞ ( M ) ✲s✉❜♠♦❞✉❧❡ ♦❢ t❤❡ C ∞ ( M ) ✲♠♦❞✉❧❡ X ( M ) ✮✳ ❊✈❛❧✉❛t✐♦♥ ❣✐✈❡s r✐s❡ t♦ ❛ ❧✐♥❡❛r ♠❛♣ ev x : X ∈ D �→ X ( x ) ∈ T x M . ■ts ✐♠❛❣❡ D x ✐s ❛ ✈❡❝t♦r s✉❜s♣❛❝❡ ♦❢ T x M ✳ ❚❤❡ ✜❡❧❞ ♦❢ ✈❡❝t♦r s♣❛❝❡s � D = D x x ∈ M ✐s ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ M ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡ ✭❣❡♥❡r❛❧❧②✱ r❛♥❦ ✈❛r②✐♥❣✮✳ ❚❤❡ ❞✐♠❡♥s✐♦♥ ♠❛♣ dim D : M → N , x �→ dim D x ✐s ❧♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s✳ ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✻ ✴ ✸✹
❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s ❋✐❜❡rs ♦❢ ❛ ❞✐str✐❜✉t✐♦♥ ❋♦r x ∈ M ✱ t❤❡ ✜❜❡r ♦❢ ( M , D ) ❛t x ✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r s♣❛❝❡✿ D x = D / I x D ✇❤❡r❡ I x = { f ∈ C ∞ ( M ) : f ( x ) = 0 } ❛♥❞ I x D ✐s t❤❡ C ∞ ( M ) ✲s✉❜♠♦❞✉❧❡ ♦❢ D ✿ I x D = { fX : f ∈ I x , X ∈ D} . ❚❤❡ ❞✐♠❡♥s✐♦♥ ♠❛♣ dim D : M → N , x �→ dim D x ✐s ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s✳ ❊✈❛❧✉❛t✐♦♥ ❣✐✈❡s r✐s❡ t♦ ❛ ❧✐♥❡❛r ♠❛♣ ev x : X ∈ D x �→ X ( x ) ∈ D x ⊂ T x M . ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✼ ✴ ✸✹
❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s ❘❛♥❦ ✈❛r②✐♥❣ ❞✐str✐❜✉t✐♦♥s ❊①❛♠♣❧❡ D ⊂ X c ( R 2 ) t❤❡ ❞✐str✐❜✉t✐♦♥ ✐♥ R 2 ❣❡♥❡r❛t❡❞ ❜② X 1 = ∂ X 2 = x ∂ ∂ x , ∂ y . ❚❤❡♥ � R 2 , ✐❢ x � = 0 , D ( x , y ) = R , ✐❢ x = 0 . D ( x , y ) = R 2 ❢♦r ❛♥② ( x , y ) ∈ R 2 . ❉❡✜♥✐t✐♦♥ ❆ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ ( M , D ) ✐s ❛ ❢❛♠✐❧② � , � D = {�· , ·� x , x ∈ M } ♦❢ ❊✉❝❧✐❞❡❛♥ ✐♥♥❡r ♣r♦❞✉❝ts �· , ·� x ♦♥ D x ✱ ✇❤✐❝❤ ✐s s♠♦♦t❤✳ ❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✽ ✴ ✸✹
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