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❘✐❡♠❛♥♥✐❛♥ 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 → Dm ⊂ TmM, dim Dm = p✳ Dm ❞❡♣❡♥❞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✐❛❧ dD : C∞(M) → C∞(M, D∗)✳ ❚❤❡ ❛ss♦❝✐❛t❡❞ ▲❛♣❧❛❝✐❛♥ ∆D = d∗

DdD : 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✿ [X, Y] = XY − YX =

• i

 

j

Xj ∂Yi ∂xj −

• j

Yj ∂Xi ∂xj   ∂ ∂xi ❢♦r X =

i Xi ∂ ∂xi , Y = i Yi ∂ ∂xi ∈ 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✉❜♠♦❞✉❧❡ ♦❢ Xc(M)✿

✶ ●✐✈❡♥ ❛♥ ♦♣❡♥ s✉❜s❡t U ♦❢ M✱ ✇❡ ♣✉t ιU : U ֒

→ M t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣✱ ❛♥❞✱ ❢♦r ❛ ✈❡❝t♦r ✜❡❧❞ X ∈ X(M)✱ ✇r✐t❡ X |U = X ◦ ιU✳ ❚❤❡ r❡str✐❝t✐♦♥ D |U ♦❢ D t♦ U ✐s t❤❡ C∞(U)✲s✉❜♠♦❞✉❧❡ ♦❢ Xc(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 X1, . . . , Xk ✐♥ X(M) s✉❝❤ t❤❛t D |U = C∞

c (U) · X1 |U + . . . + C∞ c (U) · Xk |U .

❲❡ s❛② t❤❛t t❤❡ ✈❡❝t♦r ✜❡❧❞s X1, . . . , Xk ❣❡♥❡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) ∈ Dm ∀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 ♠❛♣ evx : X ∈ D → X(x) ∈ TxM. ■ts ✐♠❛❣❡ Dx ✐s ❛ ✈❡❝t♦r s✉❜s♣❛❝❡ ♦❢ TxM✳ ❚❤❡ ✜❡❧❞ ♦❢ ✈❡❝t♦r s♣❛❝❡s D =

• x∈M

Dx ✐s ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ M ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡ ✭❣❡♥❡r❛❧❧②✱ r❛♥❦ ✈❛r②✐♥❣✮✳ ❚❤❡ ❞✐♠❡♥s✐♦♥ ♠❛♣ dimD : M → N, x → dim Dx ✐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♣❛❝❡✿ Dx = D/IxD ✇❤❡r❡ Ix = {f ∈ C∞(M) : f(x) = 0} ❛♥❞ IxD ✐s t❤❡ C∞(M)✲s✉❜♠♦❞✉❧❡ ♦❢ D✿ IxD = {fX : f ∈ Ix, X ∈ D}. ❚❤❡ ❞✐♠❡♥s✐♦♥ ♠❛♣ dimD : M → N, x → dim Dx ✐s ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s✳ ❊✈❛❧✉❛t✐♦♥ ❣✐✈❡s r✐s❡ t♦ ❛ ❧✐♥❡❛r ♠❛♣ evx : X ∈ Dx → X(x) ∈ Dx ⊂ TxM.

❨✉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 ⊂ Xc(R2) t❤❡ ❞✐str✐❜✉t✐♦♥ ✐♥ R2 ❣❡♥❡r❛t❡❞ ❜② X1 = ∂ ∂x , X2 = x ∂ ∂y . ❚❤❡♥ D(x,y) =

• R2,

✐❢ x = 0, R, ✐❢ x = 0. D(x,y) = R2 ❢♦r ❛♥② (x, y) ∈ R2. ❉❡✜♥✐t✐♦♥ ❆ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ (M, D) ✐s ❛ ❢❛♠✐❧② , D = {·, ·x, x ∈ M} ♦❢ ❊✉❝❧✐❞❡❛♥ ✐♥♥❡r ♣r♦❞✉❝ts ·, ·x ♦♥ Dx✱ ✇❤✐❝❤ ✐s s♠♦♦t❤✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✽ ✴ ✸✹

❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s

▲♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥s

❆ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❞✐str✐❜✉t✐♦♥ (M, D) ♦✈❡r ❛♥ ♦♣❡♥ s✉❜s❡t U ⊂ M ✐s ❛ ✈❡❝t♦r ❜✉♥❞❧❡ EU → U ♦✈❡r U ❡♥❞♦✇❡❞ ✇✐t❤ ❛ ♠♦r♣❤✐s♠ ♦❢ ✈❡❝t♦r ❜✉♥❞❧❡s ρU : EU → TM ♦✈❡r t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣ ιU : U → M✿ EU

• ρU TM
• U

ιU

M

s✉❝❤ t❤❛t t❤❡ ✐♥❞✉❝❡❞ ♠♦r♣❤✐s♠ ♦❢ C∞(M)✲♠♦❞✉❧❡s ρU : ΓcEU → Xc(U) ⊂ Xc(M) s❛t✐s✜❡s ρU(ΓcEU) = D |U . ■❢ σ1, . . . , σk ✐s ❛ ❢r❛♠❡ ♦❢ EU ♦✈❡r U✱ t❤❡ ♠♦❞✉❧❡ D |U ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ r❡str✐❝t✐♦♥s t♦ U ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞s Xi = ρU(σi), 1 i k.

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✾ ✴ ✸✹

❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s

▲♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥s

❆ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥✿ EU

• ρU TM
• U

ιU

M

▲❡t x ∈ U✳ ❙✐♥❝❡ ρU : ΓcEU → Xc(U) ✐s ❛ ♠♦r♣❤✐s♠ ♦❢ C∞(M)✲♠♦❞✉❧❡s✿ ρU(ΓcEU) = D |U ⇒ ρU(IxΓEU) = IxD |U t❤❡r❡❢♦r❡ ρU ✐♥❞✉❝❡s ❛ ❧✐♥❡❛r ❡♣✐♠♦r♣❤✐s♠

• ρU,x : (EU)x ∼

= ΓE/IxΓE → Dx = D |U /IxD |U .

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✵ ✴ ✸✹

❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s

❘✐❡♠❛♥♥✐❛♥ s✉❜♠❡rs✐♦♥s

(E, ·, ·E) ❛♥❞ (F, ·, ·F) ❛r❡ t✇♦ ✭✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧✮ ❊✉❝❧✐❞❡❛♥ ✈❡❝t♦r s♣❛❝❡s✳ A : E → F ✐s ❛ ❧✐♥❡❛r ❡♣✐♠♦r♣❤✐s♠✳ ❲❡ ❤❛✈❡ t❤❡ ✐♥❞✉❝❡❞ ❧✐♥❡❛r ♠❛♣ ¯ A : E/ ker A → F, ✇❤✐❝❤ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❚❤❡ ✐♥♥❡r ♣r♦❞✉❝t ·, ·E ✐♥❞✉❝❡s ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ·, ·E/ ker A ♦♥ E/ ker A✱ ✉s✐♥❣ t❤❡ ✐s♦♠♦r♣❤✐s♠ E/ ker A ∼ = (ker A)⊥✳ ❉❡✜♥✐t✐♦♥ A ✐s ❛ ❘✐❡♠❛♥♥✐❛♥ s✉❜♠❡rs✐♦♥✱ ✐❢ ¯ A ♣r❡s❡r✈❡s ✐♥♥❡r ♣r♦❞✉❝ts✿ ¯ Au, ¯ AvF = u, vE/ ker A, u, v ∈ E/ ker A.

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✶ ✴ ✸✹

❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s

❘✐❡♠❛♥♥✐❛♥ s✉❜♠❡rs✐♦♥s

■❢ A : E → F ✐s ❛ ❧✐♥❡❛r ❡♣✐♠♦r♣❤✐s♠ ❛♥❞ ·, ·E ✐s ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ E✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✐♥♥❡r ♣r♦❞✉❝t ·, ·F ♦♥ F s✉❝❤ t❤❛t A : (E, ·, ·E) → (F, ·, ·F) ✐s ❛ ❘✐❡♠❛♥♥✐❛♥ s✉❜♠❡rs✐♦♥✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦r♠ ✐s ❣✐✈❡♥ ❜② uF = ¯ A−1uE/ ker A = inf{wE : w ∈ E, Aw = u}, u ∈ F. ■❢ A : E → F ✐s ❛ ❧✐♥❡❛r ❡♣✐♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❞❥♦✐♥t A∗ : F → E ✐s ❛ ❧✐♥❡❛r ♠♦♥♦♠♦r♣❤✐s♠✳ A ✐s ❛ ❘✐❡♠❛♥♥✐❛♥ s✉❜♠❡rs✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ A∗ ✐s ❛♥ ✐s♦♠❡tr②✱ t❤❛t ✐s✱ ♣r❡s❡r✈❡s ✐♥♥❡r ♣r♦❞✉❝ts✿ A∗u, A∗vE = u, vF, ❢♦r ❛❧❧ u, v ∈ F.

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✷ ✴ ✸✹

❘✐❡♠❛♥♥✐❛♥ str✉❝t✉r❡s ♦♥ ❞✐str✐❜✉t✐♦♥s

❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ ❛ ❞✐str✐❜✉t✐♦♥

❉❡✜♥✐t✐♦♥ ❆ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ (M, D) ✐s ❛ ❢❛♠✐❧② , D = {·, ·x, x ∈ M} ♦❢ ❊✉❝❧✐❞❡❛♥ ✐♥♥❡r ♣r♦❞✉❝ts ·, ·x ♦♥ Dx✱ ✇❤✐❝❤ ✐s s♠♦♦t❤✿ ❋♦r ❡✈❡r② x ∈ M t❤❡r❡ ❡①✐st✿ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ U ♦❢ x✱ ❛ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ρU : EU → TM ♦❢ (M, D)✱ ❛ s♠♦♦t❤ ❢❛♠✐❧② {·, ·(EU)y, y ∈ U} ♦❢ ✐♥♥❡r ♣r♦❞✉❝ts ✐♥ t❤❡ ✜❜❡rs ♦❢ EU s✉❝❤ t❤❛t✱ ❢♦r ❛♥② y ∈ U✱ t❤❡ ❧✐♥❡❛r ❡♣✐♠♦r♣❤✐s♠ ˆ ρU,y : (EU)y → Dy ✐s ❛ ❘✐❡♠❛♥♥✐❛♥ s✉❜♠❡rs✐♦♥✳ ✭ρU : EU → TM ✐s ❛ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ♦❢ , D✮✳ ❚❤❡♦r❡♠ ✭■✳ ❆♥❞r♦✉❧✐❞❛❦✐s ✲ ❨✉✳❑✳✱ ✷✵✶✽✮ ❋♦r ❛♥② ❞✐str✐❜✉t✐♦♥ (M, D)✱ t❤❡r❡ ❡①✐sts ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ D✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✸ ✴ ✸✹

❚❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧

❚❤❡ ❞✉❛❧ ♦❢ ❛ ❞✐str✐❜✉t✐♦♥

❖✉r ♥❡①t ❣♦❛❧ ✐s t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧ dD : C∞(M) → C∞(M, D∗). ❍❡r❡ D∗ =

x∈M D∗ x ✐s t❤❡ ✜❡❧❞ ♦❢ t❤❡ ❞✉❛❧ s♣❛❝❡s✳

▲❡t ω∗ ❜❡ ❛ ♠❛♣ x ∈ M → ω∗(x) ∈ D∗

x✳

• ✐✈❡♥ ❛ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥ (EU, ρU)✱ ❢♦r ❛♥② x ∈ U✱ ✇❡ ❤❛✈❡ t❤❡ ❧✐♥❡❛r

❡♣✐♠♦r♣❤✐s♠ ρU,x : (EU)x → Dx✳ ❇② ❞✉❛❧✐t②✱ ✇❡ ❣❡t ❛♥ ✐♥❥❡❝t✐✈❡ ❧✐♥❡❛r ♠❛♣ ρ∗

U,x : D∗ x → E∗ U,x✳ ❚❤❡ ❧♦❝❛❧ r❡❛❧✐③❛t✐♦♥ ♦❢ ω∗ ✐s ❛ s❡❝t✐♦♥ ω∗ U ♦❢ E∗ U

❞❡✜♥❡❞ ❜② ω∗

U(y) ∈ E∗ U,y

y ∈ U

ω∗ ω∗

U

q q q q

• q

q q q

ω∗(y) ∈ D∗

y

• ρ∗

U,y

• .

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✹ ✴ ✸✹

❚❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧

❙♠♦♦t❤ s❡❝t✐♦♥s ♦❢ D∗

❉❡✜♥✐t✐♦♥ ❆ ♠❛♣ ω∗ : x ∈ M → ω∗(x) ∈ D∗

x ✐s ❛ s♠♦♦t❤ s❡❝t✐♦♥ ♦❢ D∗ ✐✛ ❢♦r ❡✈❡r②

x ∈ M t❤❡r❡ ✐s ❛ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥ (EU, ρU) ❞❡✜♥❡❞ ✐♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ x s✉❝❤ t❤❛t t❤❡ ❧♦❝❛❧ r❡❛❧✐③❛t✐♦♥ ω∗

U ♦❢ ω∗ ✐s s♠♦♦t❤ ♦♥ U✳

❆ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ s♠♦♦t❤ s❡❝t✐♦♥s ▲❡t ω∗ ❜❡ ❛ ♠❛♣ x ∈ M → ω∗(x) ∈ D∗

x✳ ❚❤❡♥ ω∗ ∈ C∞(M, D∗) ✐❢ ❛♥❞

♦♥❧② ✐❢✱ ❢♦r ❛♥② X ∈ D✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ ✐s s♠♦♦t❤ ♦♥ M✿ M ∋ x → ω∗(x), [X]x, [X]x ✐s t❤❡ ❝❧❛ss ♦❢ X ✐♥ Dx✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✺ ✴ ✸✹

❚❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧

❚❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧

C∞(M, D∗) t❤❡ s❡t ♦❢ s♠♦♦t❤ s❡❝t✐♦♥s ♦❢ D∗✳ ❊✈❛❧✉❛t✐♦♥ ❣✐✈❡s r✐s❡ t♦ ❛ ❧✐♥❡❛r ♠❛♣ evx : Dx → Dx ⊂ TxM✳ ❇② ❞✉❛❧✐t②✱ ✇❡ ❤❛✈❡ ev∗

x : T ∗ x M → D∗ x → D∗ x.

❉❡✜♥✐t✐♦♥ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧ dD : C∞(M) → C∞(M, D∗) ❞❡✜♥❡❞ ❛s dD : C∞(M)

d

− → C∞(M, T ∗M) ev∗ − → C∞(M, D∗).

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✻ ✴ ✸✹

❚❤❡ ▲❛♣❧❛❝✐❛♥

❋♦r t❤❡ ❛❞❥♦✐♥t d∗

D : C∞ c (M, D∗) → C∞ c (M)✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿

, D = {·, ·x, x ∈ M} ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ (M, D)❀ µ ❛ s♠♦♦t❤ ♣♦s✐t✐✈❡ ❞❡♥s✐t② ♦♥ M✿ ✐♥ ❧♦❝❛❧ ❝♦♦r❞✐♥❛t❡s (x1, . . . , xn)✱ µ = µ(x)|dx1 . . . dxn|, x ∈ Ω ⊂ Rn✱ ✇❤❡r❡ µ ∈ C∞(Ω)✱ µ > 0✳ ❚❤❡♥ ✇❡ ♣r♦❝❡❡❞ ❛s ❢♦❧❧♦✇s✿ ❆ ❢❛♠✐❧② , D∗ = {·, ·D∗

x , x ∈ M} ♦❢ ✐♥♥❡r ♣r♦❞✉❝ts ♦♥ D∗

x❀

❚❤❡ ♣♦✐♥t✇✐s❡ ✐♥♥❡r ♣r♦❞✉❝t ♦❢ ω, ω′ ∈ C∞(M, D∗) ✐s ❛ ❢✉♥❝t✐♦♥ ω, ω′D∗ ♦♥ M ❣✐✈❡♥ ❜② ω, ω′D∗(x) = ω(x), ω′(x)D∗

x ,

x ∈ M; ❆♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ C∞(M, D∗)✿ (ω, ω′)L2(M,D∗,µ) =

• M

ω, ω′D∗(x)dµ(x), ω, ω′ ∈ C∞(M, D∗). ❖♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ω, ω′D∗ ✐s s♠♦♦t❤✱ s♦ t❤❡ ✐♥t❡❣r❛❧ ✐s ✇❡❧❧✲❞❡✜♥❡❞✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✼ ✴ ✸✹

❚❤❡ ▲❛♣❧❛❝✐❛♥

❚❤❡ ❤♦r✐③♦♥t❛❧ ▲❛♣❧❛❝✐❛♥ ♦❢ ❛ ❞✐str✐❜✉t✐♦♥

▲❡♠♠❛ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ♦♣❡r❛t♦r d∗

D : C∞ c (M, D∗) → C∞ c (M) ✇❤✐❝❤ ✐s

❛❞❥♦✐♥t ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ ❞✐✛❡r❡♥t✐❛❧ dD✿ (dDf, ω)L2(M,D∗,µ) = (f, d∗

Dω)L2(M,µ),

u ∈ C∞

c (M),

ω ∈ C∞

c (M, D∗).

❙✐♥❝❡ D∗ ✐s ♥♦t ❛ ✈❡❝t♦r ❜✉♥❞❧❡✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❛❞❥♦✐♥t d∗

D : C∞ c (M, D∗) → C∞ c (M) ✐s ♥♦t ✐♠♠❡❞✐❛t❡✳ ❋♦r t❤❡ ♣r♦♦❢✱ ✇❡ ✉s❡ t❤❡

❛❞❥♦✐♥ts d∗

E∗

U ♦❢ t❤❡ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥s dE∗ U✳

❉❡✜♥✐t✐♦♥ ❚❤❡ ♦♣❡r❛t♦r ∆D = d∗

D ◦ dD : C∞ c (M) → C∞ c (M) ✐s ❝❛❧❧❡❞ t❤❡ ❤♦r✐③♦♥t❛❧

▲❛♣❧❛❝✐❛♥ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ (M, D)✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✽ ✴ ✸✹

❚❤❡ ▲❛♣❧❛❝✐❛♥

❙❡❧❢✲❛❞❥♦✐♥t♥❡ss

∆D ✐s ❛ ❢♦r♠❛❧❧② s❡❧❢✲❛❞❥♦✐♥t s❡❝♦♥❞ ♦r❞❡r ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ♦♥ C∞(M)✳ ❚❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ∆D ✭t❤❡ ❉✐r✐❝❤❧❡t ❢♦r♠✮✿ (∆Du, u) =

• M

|dDu(x)|2

D∗

x dµ(x),

u ∈ C∞(M). ❚❤❡♦r❡♠ ✭■✳ ❆♥❞r♦✉❧✐❞❛❦✐s ✲ ❨✉✳❑✳✱ ✷✵✶✽✮ ❚❤❡ ▲❛♣❧❛❝✐❛♥ ∆D ❝♦♥s✐❞❡r❡❞ ❛s ❛♥ ✉♥❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r ✐♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ L2(M, µ) ✇✐t❤ ❞♦♠❛✐♥ C∞(M) ✐s ❡ss❡♥t✐❛❧❧② s❡❧❢✲❛❞❥♦✐♥t ✭t❤❛t ✐s✱ ✐ts ❝❧♦s✉r❡ ∆D ✐s ❛ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r✮✳ ❚❤❡ ♣r♦♦❢ ✉s❡s ✜♥✐t❡ ♣r♦♣❛❣❛t✐♦♥ s♣❡❡❞ ❢♦r t❤❡ ❛ss♦❝✐❛t❡❞ ✇❛✈❡ ❡q✉❛t✐♦♥ ✭❈❤❡r♥♦✛✱ ✶✾✼✸✮✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✶✾ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥

❉❡✜♥✐t✐♦♥ ❆ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥ ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❆♥❞r♦✉❧✐❞❛❦✐s✲❙❦❛♥❞❛❧✐s✮ ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥ F ✇❤✐❝❤ ✐s ✐♥✈♦❧✉t✐✈❡✱ ♥❛♠❡❧② [F, F] ⊆ F✳ ■♥ ♦t❤❡r ✇♦r❞s✱ F ✐s ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜✲♠♦❞✉❧❡ F ♦❢ C∞(M; TM)✱ st❛❜❧❡ ✉♥❞❡r ❜r❛❝❦❡ts✳ ❊①❛♠♣❧❡s✿ ✶✳ ❆♥ ❛r❜✐tr❛r② ♥♦♥✲❢r❡❡ ❛❝t✐♦♥ ♦❢ ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❣r♦✉♣ ♦♥ M ❞❡✜♥❡s ❛ ❢♦❧✐❛t✐♦♥✳ ✷✳ M = R2❀ F ✐s ❣❡♥❡r❛t❡❞ ❜② X(x, y) = ∂ ∂x , Yn(x, y) =

• 0,

✐❢ x 0, x−ne− 1

x ∂

∂y ,

✐❢ x > 0, , n 0. F ✐s ♥♦t ❧♦❝❛❧❧② ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ s♦ ♥♦t ❛ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✵ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

P❛rt✐t✐♦♥ t♦ ❧❡❛✈❡s

❆♥② s✐♥❣✉❧❛r ✐♥t❡❣r❛t❡s t♦ ❛ ♣❛rt✐t✐♦♥ ♦❢ M t♦ ✐♠♠❡rs❡❞ s✉❜♠❛♥✐❢♦❧❞s ✭❧❡❛✈❡s✮ ✇✐t❤ ♥♦♥✲❝♦♥st❛♥t ❞✐♠❡♥s✐♦♥✳ ❚❤❡ ♣❛rt✐t✐♦♥ t♦ ❧❡❛✈❡s ♥♦ ❧♦♥❣❡r ❞❡t❡r♠✐♥❡s t❤❡ ❞②♥❛♠✐❝s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❝❛♥ ❜❡ ♠♦r❡ t❤❛♥ ♦♥❡ ♠♦❞✉❧❡s F ✇❤✐❝❤ ✐♥t❡❣r❛t❡ t♦ t❤❡ s❛♠❡ ♣❛rt✐t✐♦♥ t♦ ❧❡❛✈❡s✳ ✶✳ M = R❀ Fn ✐s ❣❡♥❡r❛t❡❞ ❜② xn ∂

∂x ✱ n 1✳ ❚❤r❡❡ ❧❡❛✈❡s (−∞, 0)✱ {0}✱

(0, +∞)✳ ✷✳ M = R2❀ F ✭♥♦t ❛ ❢♦❧✐❛t✐♦♥✦✮ ✐s ❣❡♥❡r❛t❡❞ ❜② X(x, y) = ∂ ∂x , Yn(x, y) =

• 0,

✐❢ x 0, x−ne− 1

x ∂

∂y ,

✐❢ x > 0, , n 0. ■♥ t❤❡ ❤❛❧❢✲♣❧❛♥❡ x < 0✱ ❧❡❛✈❡s ❛r❡ t❤❡ ❧✐♥❡s {(x, y) ∈ R2 : x < 0, y ≡ c}✳ ■♥ t❤❡ ❤❛❧❢✲♣❧❛♥❡ x > 0✱ t❤❡r❡ ✐s ❛ s✐♥❣❧❡ ❧❡❛❢ {(x, y) ∈ R2 : x > 0}✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✶ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

◆❈● ❛♥❞ ❡❧❧✐♣t✐❝ t❤❡♦r② ♦♥ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❚❤❡ ◆❈● t❡❝❤♥✐q✉❡ ♦❢ ❆✳ ❈♦♥♥❡s ✐s ❡①t❡♥❞❡❞ t♦ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s ✭t❤❡ ❤♦❧♦♥♦♠② ❣r♦✉♣♦✐❞✱ t❤❡ ❢♦❧✐❛t✐♦♥ C∗✲❛❧❣❡❜r❛s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✱ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♣s❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s✱ t❤❡ ❛♥❛❧②t✐❝ ✐♥❞❡① ♠❛♣✱ ❡❧❧✐♣t✐❝ t❤❡♦r②✮✿ ■✳ ❆♥❞r♦✉❧✐❞❛❦✐s✱ ●✳ ❙❦❛♥❞❛❧✐s✱ ❚❤❡ ❤♦❧♦♥♦♠② ❣r♦✉♣♦✐❞ ♦❢ ❛ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥✱ ❏✳ ❘❡✐♥❡ ❆♥❣❡✇✳ ▼❛t❤✳✱ ✻✷✻ ✭✷✵✵✾✮✱ ✶✕✸✼ ■✳ ❆♥❞r♦✉❧✐❞❛❦✐s✱ ●✳ ❙❦❛♥❞❛❧✐s✱ Ps❡✉❞♦❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ♦♥ ❛ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥✱ ❏✳ ◆♦♥❝♦♠♠✉t✳ ●❡♦♠✳ ✺ ✭✷✵✶✶✮✱ ✶✷✺✕✶✺✷ ❚❤❡ ▲❛♣❧❛❝✐❛♥ ∆F ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥ F ✐s ❛ ❧♦♥❣✐t✉❞✐♥❛❧❧② ❡❧❧✐♣t✐❝ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✇✐t❤ r❡s♣❡❝t t♦ F✳ ▲♦♥❣✐t✉❞✐♥❛❧ ❙♦❜♦❧❡✈ s♣❛❝❡s✿ Hs(F) = {u ∈ L2(M, µ) : (I + ∆F)s/2u ∈ L2(M, µ)} ❢♦r s 0❀ Hs(F) = H−s(F)∗ ❢♦r s < 0✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✷ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❚❤❡ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❞✐str✐❜✉t✐♦♥

▲❡t D ❜❡ ❛ ❣❡♥❡r❛❧✐s❡❞ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ ▲✐❡✲❘✐♥❡❤❛rt s✉❜❛❧❣❡❜r❛ ♦❢ (C∞(M), Xc(M)) ❛ss♦❝✐❛t❡❞ t♦ D ✐s t❤❡ C∞(M)✲s✉❜♠♦❞✉❧❡ U(D) ♦❢ Xc(M) ❣❡♥❡r❛t❡❞ ❜② ❡❧❡♠❡♥ts ♦❢ D ❛♥❞ t❤❡✐r ✐t❡r❛t❡❞ ▲✐❡ ❜r❛❝❦❡ts [X1, . . . , [Xk−1, Xk]] s✉❝❤ t❤❛t Xi ∈ D✱ i = 1, . . . , k✱ ❢♦r ❡✈❡r② k ∈ N ✭t❤❡ ♠✐♥✐♠❛❧ ▲✐❡✲❘✐♥❡❤❛rt s✉❜❛❧❣❡❜r❛ ♦❢ C∞(M, TM)✱ ✇❤✐❝❤ ❝♦♥t❛✐♥s D✮✳ ❆ss✉♠♣t✐♦♥ ❆ss✉♠❡ t❤❛t U(D) ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ C∞(M)✲♠♦❞✉❧❡✿ t❤❡r❡ ❡①✐sts X1, . . . , XN ∈ U(D) s✉❝❤ t❤❛t✱ ❢♦r ❛♥② X ∈ U(D)✱ X = a1X1 + . . . + aNXN, ai ∈ C∞(M). U(D) ✐s ❛ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥ ♦♥ M ✐♥ t❤❡ s❡♥s❡ ♦❢ ❆♥❞r♦✉❧✐❞❛❦✐s ❛♥❞ ❙❦❛♥❞❛❧✐s ✖ t❤❡ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ D✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✸ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❊①❛♠♣❧❡ ♦❢ ❛ ❜❛❞ ❞✐str✐❜✉t✐♦♥

❊①❛♠♣❧❡✿ M = R3❀ D(x,y,z) ✐s s♣❛♥♥❡❞ ❜② X(x, y, z) = ∂ ∂x , Y0(x, y, z) = ∂

∂z ,

✐❢ x 0, e− 1

x ∂

∂y + ∂ ∂z ,

✐❢ x > 0, . FD ✐s ❣❡♥❡r❛t❡❞ ❜② X(x, y, z) = ∂ ∂x , Yn(x, y, z) = ∂

∂z ,

✐❢ x 0, x−ne− 1

x ∂

∂y + ∂ ∂z ,

✐❢ x > 0, , n 0. FD ✐s ♥♦t ❛ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✹ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❙✉❜❡❧❧✐♣t✐❝ ❡st✐♠❛t❡s

❆❙❙❯▼❊✿ D ❛ s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥❀ U(D) t❤❡ ❛ss♦❝✐❛t❡❞ s✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥✳ ❚❤❡♦r❡♠ ✭■✳ ❆♥❞r♦✉❧✐❞❛❦✐s ✲ ❨✉✳❑✳✱ ✷✵✶✽❀ ❨✉✳❑✳ ✷✵✶✼✮ ❚❤❡r❡ ❡①✐sts ǫ > 0 s✉❝❤ t❤❛t✱ ❢♦r ❛♥② s ∈ R✱ u2

s+ǫ Cs

• ∆Du2

s + u2 s

• ,

u ∈ C∞(M), ✇❤❡r❡ Cs > 0 ✐s ❛ ❝♦♥st❛♥t ❛♥❞ · s ❞❡♥♦t❡s t❤❡ ♥♦r♠ ✐♥ Hs(U(D))✳ Pr♦♦❢✿ ❲❡ ❢♦❧❧♦✇ ❑♦❤♥✬s ♣r♦♦❢ ♦❢ ❍☎ ♦r♠❛♥❞❡r✬s s✉♠ ♦❢ t❤❡ sq✉❛r❡s t❤❡♦r❡♠✳ ❏✳ ❏✳ ❑♦❤♥✱ ▲❡❝t✉r❡s ♦♥ ❞❡❣❡♥❡r❛t❡ ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s✱ ✶✾✼✽ ■t ✐s ♣✉r❡❧② ❢✉♥❝t✐♦♥❛❧✲❛♥❛❧②t✐❝✦

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✺ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❍②♣♦❡❧❧✐♣t✐❝✐t②

❚❤❡♦r❡♠ ✭■✳ ❆♥❞r♦✉❧✐❞❛❦✐s ✲ ❨✉✳❑✳✱ ✷✵✶✽❀ ❨✉✳❑✳ ✷✵✶✼✮ ❋♦r s ∈ R✱ u ∈ H−∞(U(D)) :=

• t∈R

Ht(U(D)), ∆Du ∈ Hs(U(D)) ⇒ u ∈ Hs+ε(U(D)). ■♥ ♣❛rt✐❝✉❧❛r✱ u ∈ H−∞(U(D)), ∆Du ∈ H+∞(U(D)) :=

• t∈R

Ht(U(D)) ⇒ u ∈ H+∞(U(D)). ❈♦♥❝❧✉s✐♦♥✿ ❲❤❡♥ U(D) ✐s ❛ ❢♦❧✐❛t✐♦♥✱ t❤❡ ❤♦r✐③♦♥t❛❧ ▲❛♣❧❛❝✐❛♥ ∆D ✐s ❧♦♥❣✐t✉❞✐♥❛❧❧② ❤②♣♦❡❧❧✐♣t✐❝✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✻ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❚❤❡ ❣❧♦❜❛❧ s♣❡❝tr✉♠ ❛♥❞ t❤❡ ❧❡❛❢✇✐s❡ s♣❡❝tr✉♠

❆ss✉♠♣t✐♦♥ ❚❤❡ ❢♦❧✐❛t✐♦♥ U(D) ❛ss♦❝✐❛t❡❞ ✇✐t❤ D ✐s ❛ r❡❣✉❧❛r ❢♦❧✐❛t✐♦♥ F✳ ❚❤❡♦r❡♠ ✭■✳ ❆♥❞r♦✉❧✐❞❛❦✐s✲❨✳❑✳✱ ✷✵✶✾✮ ❉❡♥♦t❡ σ(∆D) ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ∆D ✐♥ L2(M, µ)❀ σF(∆D) := {σ(∆˜

L) : L ∈ M/F} ✐s ✐ts ❧❡❛❢✇✐s❡ s♣❡❝tr✉♠✱ ✇❤❡r❡

σ(∆˜

L) ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r ∆˜ L ✐♥ L2(˜

L)✳ ❚❤❡♥ ✭✶✮ σF(∆D) ⊂ σ(∆D)❀ ✭✷✮ ■❢ t❤❡ ❤♦❧♦♥♦♠② ❣r♦✉♣♦✐❞ ♦❢ F ✐s ❛♠❡♥❛❜❧❡ ✭t❤❛t ✐s✱ C∗(F) ∼ = C∗

r (F)✮✱ t❤❡♥ σ(∆D) = σF(∆D)✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✼ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

▼♦r♣❤✐s♠s ♦❢ ❍✐❧❜❡rt ♠♦❞✉❧❡s

❈♦♥s✐❞❡r t❤❡ C∗✲❛❧❣❡❜r❛ C∗(F) ❛s ❛ ❍✐❧❜❡rt ♠♦❞✉❧❡ ♦✈❡r ✐ts❡❧❢✿ t❤❡ r✐❣❤t ♠♦❞✉❧❡ str✉❝t✉r❡ ✐s ❣✐✈❡♥ ❜② t❤❡ r✐❣❤t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ❡❧❡♠❡♥ts ♦❢ t❤❡ ❛❧❣❡❜r❛✱ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ❜② a, b = a∗b✳ ❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ❛ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r PD ♦♥ t❤❡ ❤♦❧♦♥♦♠② ❣r♦✉♣♦✐❞ G✱ s✉❝❤ t❤❛t✱ ❢♦r k1, k2 ∈ C∞

c (G)✱

PD(k1 ∗ k2) = PDk1 ∗ k2. ❛♥❞ t❤❡ ♦♣❡r❛t♦r ∆D ✐s t❤❡ ✐♠❛❣❡ ♦❢ PD ✉♥❞❡r t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ Rµ ♦❢ C∗(F) ✐♥ L2(M, µ)✿ ∆DRµ(k) = Rµ(PDk), k ∈ C∞

c (G).

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✽ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❯♥❜♦✉♥❞❡❞ ♠✉❧t✐♣❧✐❡rs

❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r t❤❡ ♦♣❡r❛t♦r PD ❛s ❛♥ ✉♥❜♦✉♥❞❡❞✱ ❞❡♥s❡❧② ❞❡✜♥❡❞ ♦♣❡r❛t♦r ♦♥ t❤❡ ❍✐❧❜❡rt ♠♦❞✉❧❡ C∗(F) ✇✐t❤ ❞♦♠❛✐♥ A = C∞

c (G)✳

❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t Rµ ✐s ❛ ∗✲r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ C∗(F)✱ ✐♥❥❡❝t✐✈❡ ♦♥ C∞

c (G)✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t PD ✐s ❢♦r♠❛❧❧② s❡❧❢✲❛❞❥♦✐♥t✱ t❤❛t ✐s✱ ❢♦r

❛♥② k1, k2 ∈ C∞

c (G)✱

PDk1, k2 = k1, PDk2. ❙✐♥❝❡ PD ✐s ❞❡♥s❡❧② ❞❡✜♥❡❞✱ ✐ts ❢♦r♠❛❧ s❡❧❢✲❛❞❥♦✐♥t♥❡ss ✐♠♠❡❞✐❛t❡❧② ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❝❧♦s✉r❡ PD✳ ❚❤❡♦r❡♠ ❚❤❡ ♦♣❡r❛t♦r PD ✐s ❛♥ ✉♥❜♦✉♥❞❡❞ ♠✉❧t✐♣❧✐❡r ♦❢ C∗(F) ✭t❤❛t ✐s✱ ❛♥ ✉♥❜♦✉♥❞❡❞ r❡❣✉❧❛r ♦♣❡r❛t♦r ♦♥ t❤❡ ❍✐❧❜❡rt ♠♦❞✉❧❡ C∗(F)✮✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✷✾ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❊①❛♠♣❧❡

D t❤❡ C∞

c (R2)✲♠♦❞✉❧❡ ♦❢ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ✈❡❝t♦r ✜❡❧❞s ♦♥ R2✱

✈❛♥✐s❤✐♥❣ ❛t t❤❡ ♦r✐❣✐♥✳ ❚❤✐s ✐s t❤❡ ❢♦❧✐❛t✐♦♥ ❣❡♥❡r❛t❡❞ ❜② ✈❡❝t♦r ✜❡❧❞s X11 = x∂x, X12 = x∂y, X21 = y∂x, X22 = y∂y. D(x,y) =

• R2

✐❢ (x, y) = (0, 0); R4 ✐❢ (x, y) = (0, 0). ❋♦r α = α1(x, y)dx + α2(x, y)dy ∈ Ω1

c(R2)✱ ✇❡ ❤❛✈❡

ev∗(α)(x, y) =

• (α1(x, y), α2(x, y)) ∈ D∗

(x,y) ∼

= R2 ✐❢ (x, y) = (0, 0), 0 ∈ D∗

(0,0) ∼

= R4 ✐❢ (x, y) = (0, 0). ❋♦r f ∈ C∞

c (R2)✱

dDf(x, y) =

• (∂xf(x, y), ∂yf(x, y)) ∈ D∗

(x,y) ∼

= R2 ✐❢ (x, y) = (0, 0), 0 ∈ D(0,0) ∼ = R4 ✐❢ (x, y) = (0, 0).

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✸✵ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❊①❛♠♣❧❡

❆ ❧♦❝❛❧ ♣r❡s❡♥t❛t✐♦♥ EU ♥❡❛r (0, 0) ✐s ❣✐✈❡♥ ❜②✿ EU = R2 × R4 t❤❡ tr✐✈✐❛❧ ✈❡❝t♦r ❜✉♥❞❧❡ ♦✈❡r U = R2❀ ■❢ ✇❡ ❞❡♥♦t❡ ❜② {σij, i, j = 1, 2} t❤❡ st❛♥❞❛r❞ ❜❛s❡ ✐♥ R4 ❛♥❞ ❜② {σ∗

ij, i, j = 1, 2} t❤❡ ❞✉❛❧ ❜❛s❡ ✐♥ (R4)∗✱ t❤❡♥

ρU : EU = R2 × R4 → TR2 = R2 × R2, σij → Xij, ✇❤❡r❡ X11 = x∂x, X12 = x∂y, X21 = y∂x, X22 = y∂y.

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✸✶ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❊①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ ♠❛♣ α : R2 ∋ (x, y) → α(x, y) ∈ D∗

(x,y) =

• R2

✐❢ (x, y) = (0, 0); R4 ✐❢ (x, y) = (0, 0). . ❇② ❞❡✜♥✐t✐♦♥✱ α ✐s ❛ s♠♦♦t❤ s❡❝t✐♦♥ ♦❢ D∗ ✐✛ ✐ts ❧♦❝❛❧ r❡❛❧✐③❛t✐♦♥ αU ❞❡✜♥❡❞ ❜② αU = ρ∗

U ◦ α ✐s s♠♦♦t❤ ♦♥ R2✳

■❢ ✇❡ ✇r✐t❡ α ♦♥ R2 \ {0} ❛s α = α1(x, y)dx + α2(x, y)dy✱ t❤❡♥

• ρ∗

Uα = xα1σ∗ 11 + xα2σ∗ 12 + yα1σ∗ 21 + yα2σ∗ 22,

α ✐s s♠♦♦t❤ ✐✛ xα1, xα2, yα1, yα2 ❛r❡ ❡①t❡♥❞❡❞ t♦ s♠♦♦t❤ ❢✉♥❝t✐♦♥s ♦♥ R2✳

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✸✷ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❊①❛♠♣❧❡

❚❤❡ r❡str✐❝t✐♦♥ ♦❢ ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ D t♦ R2 \ {0} ✐s ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❡tr✐❝ ♦♥ t❤❡ ♠❛♥✐❢♦❧❞ R2 \ {0}✿ g(x,y) = A(x, y)dx2 + 2B(x, y)dx dy + C(x, y)dy2, (x, y) = (0, 0). ✇✐t❤ s♦♠❡ A, B, C ∈ C∞

c (R2 \ {0}).

▲❡t {G(x,y), (x, y) ∈ R2} ❜❡ ❛ s♠♦♦t❤ ❢❛♠✐❧② ♦❢ ✐♥♥❡r ♣r♦❞✉❝ts ✐♥ t❤❡ ✜❜❡rs ♦❢ EU✿ G(x,y) =

• i1,j1,i2,j2=1,2

Gi1j1,i2j2(x, y)σ∗

i1j1σ∗ i2j2.

❋♦r ❛♥② (x, y) ∈ R2✱ t❤❡ ♠❛♣ ρU,(x,y) : R4 → R2 ✐s ❛ ❘✐❡♠❛♥♥✐❛♥ s✉❜♠❡rs✐♦♥✱ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ ρ∗

U,(x,y) : (R2)∗ ∼

= T ∗

(x,y)R2 → (R4)∗ ✐s ❛♥

✐s♦♠❡tr②✳ ❋♦r α = α1(x, y)dx + α2(x, y)dy ∈ Ω1

c(R2)✱ ✇❡ ❤❛✈❡

α(x, y)2

g−1 = ρ∗ Uα(x, y)2 G−1.

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✸✸ ✴ ✸✹

❙✐♥❣✉❧❛r ❢♦❧✐❛t✐♦♥s

❊①❛♠♣❧❡

■❢ G ✐s t❤❡ st❛♥❞❛r❞ ♠❡tr✐❝ ♦♥ R4✱ t❤❡♥ (σij, i, j = 1, 2) ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s❡ ✐♥ R4 ❛♥❞ α(x, y)2

g−1 = ρ∗ Uα(x, y)2 G−1 = (x2 + y2)(α2 1(x, y) + α2 2(x, y)).

❲❡ ❣❡t g(x,y) = 1 x2 + y2 (dx2 + dy2), (x, y) = (0, 0). ❚❤❡ ♣♦s✐t✐✈❡ ❞❡♥s✐t② µ ♦♥ R2 ✐s ❣✐✈❡♥ ❜② µ = dx dy✳ ❋♦r α ∈ C∞

c (R2, D∗) ♦❢ t❤❡ ❢♦r♠ α = α1(x, y)dx + α2(x, y)dy ♦♥

R2 \ {0}✱ d∗

Dα ∈ C∞ c (R2, D∗) ✐s ❣✐✈❡♥ ❜②

d∗

Dα(x, y) = − ∂

∂x ((x2 + y2)α1) − ∂ ∂y ((x2 + y2)α2). ❋♦r f ∈ C∞

c (R2)✱ ∆Df ∈ C∞ c (R2) ✐s ❣✐✈❡♥ ❜②

∆Df(x, y) = − ∂ ∂x

• (x2 + y2) ∂f

∂x

• − ∂

∂y

• (x2 + y2) ∂f

∂y

• .

❨✉r✐ ❆✳ ❑♦r❞②✉❦♦✈ ✭❯❢❛✱ ❘✉ss✐❛✮ ▲❛♣❧❛❝✐❛♥s ❢♦r s♠♦♦t❤ ❞✐str✐❜✉t✐♦♥s P♦ts❞❛♠✱ ✷✵✶✾ ✸✹ ✴ ✸✹