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❲♦r❦s❤♦♣ ♦♥ ●❡♦♠❡tr② ▼❡t❤♦❞s ✐♥ P❤②s✐❝s

❇✐❛➟♦✇✐❡➺❛ ✷✵✶✾

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ G✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

• r③❡❣♦r③ ❏❛❦✐♠♦✇✐❝③

■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ✐♥ ❇✐❛➟②st♦❦

❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥❡t❛ ❙❧✐➺❡✇s❦❛ ❛♥❞ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ✒❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ G✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s✧✱ ❛r❳✐✈✿✶✼✶✷✳✵✽✹✷✵ t♦ ❛♣♣❡❛r ✐♥ ❘❡✈✐❡✇s ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❚❤❡ ♣❤❛s❡ s♣❛❝❡ ♦❢ ❛ t②♣✐❝❛❧ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ✐s t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ T ∗P ♦❢ ✐ts ❝♦♥✜❣✉r❛t✐♦♥s s♣❛❝❡ P ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠✳ ❯s✉❛❧❧② ♦♥❡ ❝♦♥s✐❞❡rs t❤❡ ❝❛s❡ ✇❤❡♥ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ t❤✐s s②st❡♠ ✐s ✐♥✈❛r✐❛♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦t❛♥❣❡♥t ❧✐❢t ♦❢ ❛♥ ❛❝t✐♦♥ ♦❢ s♦♠❡ ❣r♦✉♣ G ♦♥ P✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s s♦♠❡t✐♠❡s r❡❛s♦♥❛❜❧❡ t♦ r❡♣❧❛❝❡ t❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠ ✇✐t❤ ❛♥♦t❤❡r G✲✐♥✈❛r✐❛♥t s②♠♣❧❡❝t✐❝ ❢♦r♠ ♦♥ T ∗P ✇❤✐❝❤ r❡t❛✐♥s ❝❡rt❛✐♥ ♣r♦♣❡rt✐❡s✳ ❆ss✉♠✐♥❣ t❤❛t t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ P ✐s ❢r❡❡ ❛♥❞ t❤❡ q✉♦t✐❡♥t s♣❛❝❡ P/G ✐s ❛ ♠❛♥✐❢♦❧❞ M ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r P ❛s t❤❡ t♦t❛❧ s♣❛❝❡ ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ G✲❜✉♥❞❧❡ P(M, G)✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P(M, G) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ ❚❤❡ ❣r♦✉♣ ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ ✳ ❚❤❡ ❣r♦✉♣ ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ ❚❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡ ✳ ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P(M, G) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳

• ❚❤❡ ❣r♦✉♣ AutTGTP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢

t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP✳ ❚❤❡ ❣r♦✉♣ ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ ❚❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡ ✳ ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P(M, G) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳

• ❚❤❡ ❣r♦✉♣ AutTGTP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢

t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP✳ ❚❤❡ ❣r♦✉♣ AutTGTP ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿ ❚❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ✲❜✉♥❞❧❡ ✳ ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P(M, G) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳

• ❚❤❡ ❣r♦✉♣ AutTGTP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢

t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP✳ ❚❤❡ ❣r♦✉♣ AutTGTP ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿

• ❚❤❡ s♣❛❝❡ ConnP(M, G) ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧

G✲❜✉♥❞❧❡ P(M, G)✳ ❚❤❡ s♣❛❝❡ ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s ♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ ✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ P(M, G) ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳

• ❚❤❡ ❣r♦✉♣ AutTGTP ✭❛♥❞ ✐ts s✉❜❣r♦✉♣s✮ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢

t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ P ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ❛❝t✐♦♥ ♦❢ TG ♦♥ TP✳ ❚❤❡ ❣r♦✉♣ AutTGTP ❛❝ts ♦♥ t❤❡ ❜♦t❤ s♣❛❝❡s✿

• ❚❤❡ s♣❛❝❡ ConnP(M, G) ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧

G✲❜✉♥❞❧❡ P(M, G)✳

• ❚❤❡ s♣❛❝❡ CanT ∗P ♦❢ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠s γ

♦♥ t❤❡ ❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ T ∗P✱ ✇❤✐❝❤ ❛♥♥✐❤✐❧❛t❡ t❤❡ ✈❡❝t♦rs t❛♥❣❡♥t t♦ t❤❡ ✜❜r❡s ♦❢ T ∗P✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

G✲♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡

• G✲♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡ ♦✈❡r ❛ ♠❛♥✐❢♦❧❞ M

G P M ∼ = P/G

❄ ✲

µ ✇❤❡r❡ t❤❡ ❢r❡❡ ❛❝t✐♦♥ ♦❢ G ✇❡ ❞❡♥♦t❡ ❜② κ : P × G → P, κ(p, g) := pg ❛♥❞ κg : P → P κg(p) := pg κp : G → P κp(g) := pg

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❆❝t✐♦♥ ♦❢ TG ♦♥ TP

• TG ✐s ❛ ▲✐❡ ❣r♦✉♣ ✇✐t❤ t❤❡ ♣r♦❞✉❝t ❛♥❞ t❤❡ ✐♥✈❡rs❡ ❞❡✜♥❡❞ ❜②

t❤❡ t❛♥❣❡♥t t❤❡ ♣r♦❞✉❝t m ❛♥❞ t♦ t❤❡ ✐♥✈❡rs❡ ι ✐♥ ●✿ Tm(g,h)(Xg, Yh) =: Xg • Yh = TLg(h)Yh + TRh(g)Xg, ✭✶✮ Tιg(Xg) =: X−1

g

= −TLg−1(e) ◦ TRg−1(g)Xg ✭✷✮ ✇❤❡r❡ Xg ∈ TgG✱ Yh ∈ ThG ❛♥❞ Lg(h) := gh✱ Rg(h) := hg✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❆❝t✐♦♥ ♦❢ TG ♦♥ TP

❚❤❡ ❞✐✛❡♦♠♦r♣❤✐s♠ I : G × TeG → TG I(g, Xe) = TRg(e)Xe =: Xg ✭✸✮ ❛❧❧♦✇s ✉s t♦ ❝♦♥s✐❞❡r TG ❛s t❤❡ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t G ⋉AdG TeG ♦❢ G ❜② t❤❡ TeG✱ ✇❤❡r❡ t❤❡ ❣r♦✉♣ ♣r♦❞✉❝t ♦❢ (g, Xe), (h, Ye) ∈ G ⋉AdG TeG ✐s ❣✐✈❡♥ ❜② (g, Xe) • (h, Ye) = I−1(I(g, Xe) · I(h, Ye)) = ✭✹✮ = (gh, Xe + T(Rg−1 ◦ Lg)(e)Ye) = (gh, Xe + AdgYe).

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❆❝t✐♦♥ ♦❢ TG ♦♥ TP

❯s✐♥❣ t❤❡ ❛❜♦✈❡ ✐s♦♠♦r♣❤✐s♠s ✇❡ ♦❜t❛✐♥ t❤❡ ❛❝t✐♦♥ ♦❢ G ⋉AdG TeG ♦♥ t❤❡ t❛♥❣❡♥t ❜✉♥❞❧❡ TP ❛s t❤❡ t❛♥❣❡♥t t♦ κ Φ(g,Xe)(vp) := Tκg,p((g, Xe), vp) = Tκg(p)vp + Tκp(g)TRg(e)Xe ✭✺✮ ✳ ❆♣♣❧②✐♥❣ t❤❡ ❛❜♦✈❡ ❛❝t✐♦♥ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s♦♠♦r♣❤✐s♠s TP/T vP ∼ = TP/TeG, ✭✻✮ TP/TG ∼ = (TP/TeG)/G ∼ = (TP/G)/TeG, ✭✼✮ TM = T(P/G) ∼ = TP/TG, ✭✽✮ ♦❢ ✈❡❝t♦r ❜✉♥❞❧❡s✱ ✇❤❡r❡ ✇❡ ✇r✐t❡ T vP := KerTµ ❢♦r t❤❡ ✈❡rt✐❝❛❧ s✉❜❜✉♥❞❧❡ ♦❢ TP✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

• r♦✉♣s ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ TP
• ❲❡ ❝♦♥s✐❞❡r t❤❡ ❣r♦✉♣ Aut0(TP) ♦❢ s♠♦♦t❤ ❛✉t♦♠♦r♣❤✐s♠s

TP TP P P

❄ ❄ ✲ ✲

π π A id A(p) : TpP → TpP ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ t❛♥❣❡♥t s♣❛❝❡ TpP ❞❡♣❡♥❞s s♠♦♦t❤❧② ♦♥ p

• Aut0(TP) ✐s ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣ ♦❢ t❤❡ ❣r♦✉♣ Aut(TP) ♦❢ ❛❧❧

❛✉t♦♠♦r♣❤✐s♠s ♦❢ TP✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

• r♦✉♣s ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ TP
• ❚❤❡ s✉❜❣r♦✉♣ AutTG(TP) ⊂ Aut0(TP) ❝♦♥s✐st✐♥❣ ♦❢ t❤♦s❡

❡❧❡♠❡♥ts ♦❢ Aut0(TP) ✇❤♦s❡ ❛❝t✐♦♥ ♦♥ TP ❝♦♠♠✉t❡s ✇✐t❤ t❤❡ ❛❝t✐♦♥ ✭✺✮ ♦❢ TG ∼ = G ⋉Adg TeG ♦♥ TP✱ ✐✳❡✳ TP TP TP TP

❄ ❄ ✲ ✲

Φ Φ A A A(pg) ◦ Φ(g,Xe) = Φ(g,Xe) ◦ A(p)

• ❚❤❡ ❣r♦✉♣ AutTG(TP) ❛❝ts ❛❧s♦ ♦♥ ✈❡❝t♦r ❜✉♥❞❧❡s TP/G → M

❛♥❞ TM → M✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

• r♦✉♣s ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ TP
• ❲❡ ❞❡✜♥❡ t❤❡ s✉❜❣r♦✉♣ AutNTP ⊂ AutTGTP ❝♦♥s✐st✐♥❣ ♦❢

A ∈ AutTGTP s✉❝❤ t❤❛t A(p) = idp + B(p)✱ ✇❤❡r❡ B(p) : TpP → T v

p P✳

❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ B(p) ♦♥❡ ❤❛s ImB(p) ⊂ T v

p P ⊂ KerB(p)✳

❚❤✉s B1(p)B2(p) = 0 ❢♦r ❛♥② id + B1, id + B2 ∈ AutNP✳ ❙♦✱ ♦♥❡ ❤❛s (idp + B1(p))(idp + B2(p)) = idp + B1(p) + B2(p). ✭✾✮ ❚❤✐s s❤♦✇s t❤❛t AutNTP ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ s✉❜❣r♦✉♣ ♦❢ AutTGTP✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ❖♥❡ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ {0} → AutNTP

ι

→ AutTGTP

λ

→ Aut0TM → {id} ✭✶✵✮ ♦❢ t❤❡ ❣r♦✉♣ ♠♦r♣❤✐s♠s✱ ✇❤❡r❡ ι ✐s t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣ ❛♥❞ λ ✐s ❛♥ ❡♣✐♠♦r♣❤✐s♠ ❝♦✈❡r✐♥❣ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ M ❞❡✜♥❡❞ ❜② λ(A)(µ(p))(Tµ(p))vp := (Tµ(p) ◦ A(p))vp ✭✶✶✮ ❢♦r vp ∈ TpP✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❈♦♥♥❡❝t✐♦♥ ❢♦r♠

• ❆ ❝♦♥♥❡❝t✐♦♥ ❢♦r♠ ♦♥ P ✐s ❛ TeG✲✈❛❧✉❡❞ ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠ α

s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s αp ◦ Tκp(e) = idTeG ✭✶✷✮ αpg ◦ Tκg(p) = Adg−1 ◦ αp ✭✶✸✮ ✈❛❧✐❞ ❢♦r ✈❛❧✉❡ αp ♦❢ α ❛t p ∈ P ❛♥❞ g ∈ G✳ ❯s✐♥❣ α ♦♥❡ ❞❡✜♥❡s t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ TpP = T v

p P ⊕ T α,h p

P ✭✶✹✮ ♦❢ TpP ♦♥ t❤❡ ✈❡rt✐❝❛❧ T v

p P ❛♥❞ t❤❡ ❤♦r✐③♦♥t❛❧ T α,h p

P := Kerαp s✉❜s♣❛❝❡s ✇❤✐❝❤ ❛❧s♦ s❛t✐s❢② t❤❡ G✲❡q✉✐✈❛r✐❛♥❝❡ ♣r♦♣❡rt✐❡s Tκg(p)T v

p P = T v pgP,

✭✶✺✮ Tκg(p)T α,h

p

P = T α,h

pg P.

✭✶✻✮

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❋r♦♠ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✶✹✮ ❢♦r ❛♥② p ∈ P ♦♥❡ ♦❜t❛✐♥s t❤❡ ✈❡❝t♦r s♣❛❝❡s ✐s♦♠♦r♣❤✐s♠ Γα(p) : Tµ(p)M → T α,h

p

P ✭✶✼✮ s✉❝❤ t❤❛t Γα(pg) = Tκg(p) ◦ Γα(p) ✭✶✽✮ ❛♥❞ Tµ(p) ◦ Γα(p) = idµ(p), Γα(p) ◦ Tµ(p) = Πh

α(p),

✭✶✾✮ ✇❤❡r❡ Πh

α(p) ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥

idp = Πv

α(p) + Πh α(p)

✭✷✵✮ ♦❢ t❤❡ ✐❞❡♥t✐t② ♠❛♣ ♦❢ TpP ♦♥ t❤❡ ♣r♦❥❡❝t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✭✶✹✮✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ❆ ✜①❡❞ ❝♦♥♥❡❝t✐♦♥ α ❞❡✜♥❡s t❤❡ ✐♥❥❡❝t✐♦♥ σα : Aut0TM → AutTGTP σα( ˜ A)(p) := Πv

α(p) + Γα(p) ◦ ˜

A(µ(p)) ◦ Tµ(p), ✭✷✶✮ ✇❤❡r❡ ˜ A ∈ Aut0TM✱ t❤❡ s✉r❥❡❝t✐♦♥ βα : AutTGTP → AutNTP ❜② βα(A) := Aσα(λ(A))−1, ✇❤❡r❡ A ∈ AutTGTP✱ ✇❤✐❝❤ ❛r❡ ❛rr❛♥❣❡❞ ✐♥t♦ t❤❡ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ {✐❞TM} Aut0TM AutTGTP AutNTP {idTP }

✲ ✲ ✲ ✲

σα βα ✐♥✈❡rs❡ t♦ t❤❡ s❡q✉❡♥❝❡ ✭✶✵✮✳ ❚❤❡ ♠❛♣ ✐s ❛ ♠♦♥♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❣r♦✉♣s ❛♥❞ s❛t✐s✜❡s

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ❆ ✜①❡❞ ❝♦♥♥❡❝t✐♦♥ α ❞❡✜♥❡s t❤❡ ✐♥❥❡❝t✐♦♥ σα : Aut0TM → AutTGTP σα( ˜ A)(p) := Πv

α(p) + Γα(p) ◦ ˜

A(µ(p)) ◦ Tµ(p), ✭✷✶✮ ✇❤❡r❡ ˜ A ∈ Aut0TM✱ t❤❡ s✉r❥❡❝t✐♦♥ βα : AutTGTP → AutNTP ❜② βα(A) := Aσα(λ(A))−1, ✇❤❡r❡ A ∈ AutTGTP✱ ✇❤✐❝❤ ❛r❡ ❛rr❛♥❣❡❞ ✐♥t♦ t❤❡ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ {✐❞TM} Aut0TM AutTGTP AutNTP {idTP }

✲ ✲ ✲ ✲

σα βα

✛ ✛

λ ι ✐♥✈❡rs❡ t♦ t❤❡ s❡q✉❡♥❝❡ ✭✶✵✮✳ ❚❤❡ ♠❛♣ ✐s ❛ ♠♦♥♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❣r♦✉♣s ❛♥❞ s❛t✐s✜❡s

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ❆ ✜①❡❞ ❝♦♥♥❡❝t✐♦♥ α ❞❡✜♥❡s t❤❡ ✐♥❥❡❝t✐♦♥ σα : Aut0TM → AutTGTP σα( ˜ A)(p) := Πv

α(p) + Γα(p) ◦ ˜

A(µ(p)) ◦ Tµ(p), ✭✷✶✮ ✇❤❡r❡ ˜ A ∈ Aut0TM✱ t❤❡ s✉r❥❡❝t✐♦♥ βα : AutTGTP → AutNTP ❜② βα(A) := Aσα(λ(A))−1, ✇❤❡r❡ A ∈ AutTGTP✱ ✇❤✐❝❤ ❛r❡ ❛rr❛♥❣❡❞ ✐♥t♦ t❤❡ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ {✐❞TM} Aut0TM AutTGTP AutNTP {idTP }

✲ ✲ ✲ ✲

σα βα

✛ ✛

λ ι ✐♥✈❡rs❡ t♦ t❤❡ s❡q✉❡♥❝❡ ✭✶✵✮✳ ❚❤❡ ♠❛♣ σα ✐s ❛ ♠♦♥♦♠♦r♣❤✐s♠ σα( ˜ A1 ˜ A2) = σα( ˜ A1)σα( ˜ A2) ♦❢ t❤❡ ❣r♦✉♣s ❛♥❞ βα s❛t✐s✜❡s βα(A1A2) = βα(A1)σα(λ(A1))βα(A2)σα(λ(A1))−1.

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ❯s✐♥❣ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ A(p) = (✐❞p + B(p))σα( ˜ A)(p) ✭✷✷✮ ♦❢ A ∈ AutTGTP✱ ✇❤❡r❡ ✐❞p + B(p) ∈ AutNTP ❛♥❞ ˜ A ∈ Aut0TM✱ ♦♥❡ ❞❡✜♥❡s ❛♥ ✐s♦♠♦r♣❤✐s♠ AutTGTP − → Aut0TM ⋉α EndNTP, ✇❤❡r❡ t❤❡ ♣r♦❞✉❝t ♦❢ ✐s ❣✐✈❡♥ ❜② ✭✷✸✮

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ❯s✐♥❣ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ A(p) = (✐❞p + B(p))σα( ˜ A)(p) ✭✷✷✮ ♦❢ A ∈ AutTGTP✱ ✇❤❡r❡ ✐❞p + B(p) ∈ AutNTP ❛♥❞ ˜ A ∈ Aut0TM✱ ♦♥❡ ❞❡✜♥❡s ❛♥ ✐s♦♠♦r♣❤✐s♠ AutTGTP − → Aut0TM ⋉α EndNTP, ✇❤❡r❡ t❤❡ ♣r♦❞✉❝t ♦❢ ( ˜ A1, B1), ( ˜ A2, B2) ∈ Aut0TM ⋉α EndNTP ✐s ❣✐✈❡♥ ❜② [( ˜ A1, B1) · ( ˜ A2, B2)](p) := ✭✷✸✮ = ( ˜ A1(µ(p)) ˜ A2(µ(p)), B1(p)+B2(p)◦Γα(p)◦ ˜ A−1

1 (µ(p))◦Tµ(p)).

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❆❝t✐♦♥ ♦♥ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s

• ▲❡t ConnP(M, G) ❜❡ t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❝♦♥♥❡❝t✐♦♥s ♦♥ P(M, G)✳

❲❡ ❞❡✜♥❡ φA(α)p := αp ◦ A(p)−1 ✭✷✹✮ t❤❡ ❧❡❢t ❛❝t✐♦♥ φA : ConnP(M, G) → ConnP(M, G) ♦❢ AutTGTP ♦♥ ConnP(M, G)✱ ✐✳❡✳ φ s❛t✐s✜❡s φA1A2 = φA1 ◦ φA1 ❢♦r A1, A2 ∈ AutTGTP✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ s❤♦✇s t❤❛t ♦♥❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❣r♦✉♣ AutTGTP ✐♥ t❡r♠s ♦❢ ❝♦♥♥❡❝t✐♦♥s s♣❛❝❡ ConnP(M, G)✳ Pr♦♣♦s✐t✐♦♥ ■❢ A ∈ Aut0(TP) ❛♥❞ φA(ConnP(M, G)) ⊂ ConnP(M, G) t❤❡♥ A ∈ AutTG(TP)✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❚❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠

• ❲❡ r❡❝❛❧❧ t❤❛t t❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠ ♦♥ T ∗P ✐s ω0 = dγ0✱

✇❤❡r❡ γ0 ∈ C∞T ∗(T ∗P) ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ♦♥❡✲❢♦r♠ ♦♥ T ∗P ❞❡✜♥❡❞ ❛t ϕ ∈ T ∗P ❜② γ0ϕ, ξϕ := ϕ, Tπ∗(ϕ)ξϕ, ✇❤❡r❡ π∗ : T ∗P → P ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ T ∗P ♦♥ t❤❡ ❜❛s❡ ❛♥❞ ξϕ ∈ Tϕ(T ∗P)✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❆ ❧✐♥❡❛r ✈❡❝t♦r ✜❡❧❞

• ❇② ❞❡✜♥✐t✐♦♥ ❛ ❧✐♥❡❛r ✈❡❝t♦r ✜❡❧❞ ♦♥ T ∗P ✐s ❛ ♣❛✐r (ξ, χ) ♦❢ ✈❡❝t♦r

✜❡❧❞s ξ ∈ C∞T(T ∗P) ❛♥❞ χ ∈ C∞TP s✉❝❤ t❤❛t T ∗P T(T ∗P) P TP

❄ ❄ ✲ ✲

π∗ Tπ∗ ξ χ ❞❡✜♥❡s ❛ ♠♦r♣❤✐s♠ ♦❢ ✈❡❝t♦r ❜✉♥❞❧❡s✳ ◆♦t❡ ❤❡r❡ t❤❛t Tπ∗(ϕ)ξϕ = χπ∗(ϕ)✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❆ ❧✐♥❡❛r ✈❡❝t♦r ✜❡❧❞

• ❲❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② LinC∞T(T ∗P) t❤❡ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ ❧✐♥❡❛r ✈❡❝t♦r

✜❡❧❞s ♦✈❡r t❤❡ ✈❡❝t♦r ❜✉♥❞❧❡ π∗ : T ∗P → P✳ ❚❤❡ ▲✐❡ ❜r❛❝❦❡t ♦❢ (ξ1, χ1)✱ (ξ2, χ2) ∈ LinC∞T(T ∗P) ✐s ❞❡✜♥❡❞ ❜② [(ξ1, χ1), (ξ2, χ2)] := ([ξ1, ξ2], [χ1, χ2]) ❛♥❞ t❤❡ ✈❡❝t♦r s♣❛❝❡ str✉❝t✉r❡ ♦♥ LinC∞T(T ∗P) ❜② c1(ξ1, χ1) + c2(ξ2, χ2) := (c1ξ1 + c2ξ2, c1χ1 + c2χ2). ▲❡t LinC∞(T ∗P) ❞❡♥♦t❡ t❤❡ ✈❡❝t♦r s♣❛❝❡ ♦❢ s♠♦♦t❤ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦♥ T ∗P✳ ❙♣❛❝❡s LinC∞T(T ∗P) ❛♥❞ LinC∞(T ∗P) ❤❛✈❡ str✉❝t✉r❡s ♦❢ C∞(P)✲♠♦❞✉❧❡s ❞❡✜♥❡❞ ❜② f(ξ, χ) := ((f ◦ π∗)ξ, fχ) ❛♥❞ ❜② fl := (f ◦ π∗)l✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡r❡ f ∈ C∞(P) ❛♥❞ l ∈ LinC∞(T ∗P)✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❉❡✜♥✐t✐♦♥

• ❆ ❞✐✛❡r❡♥t✐❛❧ ♦♥❡✲❢♦r♠ γ ∈ C∞T ∗(T ∗P) ✐s ❝❛❧❧❡❞ ❛ ❣❡♥❡r❛❧✐③❡❞

❝❛♥♦♥✐❝❛❧ ❢♦r♠ ♦♥ T ∗P ✐❢✿ ✭✐✮ γϕ = 0 ❢♦r ❛♥② ϕ ∈ T ∗P✱ ✭✐✐✮ kerTπ∗(ϕ) ⊂ ker γϕ ✭✐✐✐✮ γ, ξ ∈ LinC∞(T ∗P) ❢♦r ❛♥② ξ ∈ LinC∞T(T ∗P)✳ ❚❤❡ s♣❛❝❡ ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❝❛♥♦♥✐❝❛❧ ❢♦r♠s ♦♥ T ∗P ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② CanT ∗P✳ ▲❡t ✉s ♥♦t❡ ❤❡r❡ t❤❛t γ0 ∈ CanT ∗P✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ✭✐✮ ❚❤❡ ♠❛♣ Θ : Aut0TP → CanT ∗P ❞❡✜♥❡❞ ❜② Θ(A)ϕ, ξϕ := ϕ, A(π∗(ϕ))Tπ∗(ϕ))ξϕ, ✭✷✺✮ ✇❤❡r❡ ξϕ ∈ Tϕ(T ∗P)✱ ✐s ❜✐❥❡❝t✐✈❡✳ ✭✐✐✮ ❚❤❡ ♥❛t✉r❛❧ ❧❡❢t ❛❝t✐♦♥ ♦❢ ♦♥ ❞❡✜♥❡❞ ❜② ✭✷✻✮ ✇❤❡r❡ ✐s t❤❡ ❞✉❛❧ ♦❢ ✱ ✐s ❛ tr❛♥s✐t✐✈❡ ❛♥❞ ❢r❡❡ ❛❝t✐♦♥✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ✭✐✮ ❚❤❡ ♠❛♣ Θ : Aut0TP → CanT ∗P ❞❡✜♥❡❞ ❜② Θ(A)ϕ, ξϕ := ϕ, A(π∗(ϕ))Tπ∗(ϕ))ξϕ, ✭✷✺✮ ✇❤❡r❡ ξϕ ∈ Tϕ(T ∗P)✱ ✐s ❜✐❥❡❝t✐✈❡✳ ✭✐✐✮ ❚❤❡ ♥❛t✉r❛❧ ❧❡❢t ❛❝t✐♦♥ L∗ : Aut0TP × CanT ∗P → CanT ∗P ♦❢ Aut0TP ♦♥ CanT ∗P ❞❡✜♥❡❞ ❜② (L∗

A(γ))ϕ, ξϕ := γA∗(ϕ), TA∗(ϕ)ξϕ,

✭✷✻✮ ✇❤❡r❡ A∗ : T ∗P → T ∗P ✐s t❤❡ ❞✉❛❧ ♦❢ A ∈ Aut0TP✱ ✐s ❛ tr❛♥s✐t✐✈❡ ❛♥❞ ❢r❡❡ ❛❝t✐♦♥✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❚❤❡ ❧✐❢t Φ∗

g : T ∗P → T ∗P ♦❢ t❤❡ ❛❝t✐♦♥ κg : P → P t♦ t❤❡

❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ T ∗P ✐s ❞❡✜♥❡❞ ❜② Φ∗

g(ϕ)(pg) = (Tκg(p)−1)∗ϕ

✭✷✼✮ ✇❤❡r❡ p = π∗(ϕ)✳ ❋♦r G✲✐♥✈❛r✐❛♥t s②♠♣❧❡❝t✐❝ ❢♦r♠ ωA = dγ = dΘ(A) ♦♥❡ ❤❛s t❤❡ G✲❡q✉✐✈❛r✐❛♥t ♠♦♠❡♥t✉♠ ♠❛♣ JA : T ∗P → T ∗

e G ❣✐✈❡♥ ❜②

JA = J0 ◦ A∗. ❋♦r t❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠ t❤❡ ♠♦♠❡♥t✉♠ ♠❛♣ ✐s ■t ✐s r❡❛s♦♥❛❜❧❡ t♦ ❞❡✜♥❡ t❤❡ s♣❛❝❡ ✇❤✐❝❤ ✐s ❛♥ ✲✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡ ♦❢ t❤❡ s♣❛❝❡ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❚❤❡ ❧✐❢t Φ∗

g : T ∗P → T ∗P ♦❢ t❤❡ ❛❝t✐♦♥ κg : P → P t♦ t❤❡

❝♦t❛♥❣❡♥t ❜✉♥❞❧❡ T ∗P ✐s ❞❡✜♥❡❞ ❜② Φ∗

g(ϕ)(pg) = (Tκg(p)−1)∗ϕ

✭✷✼✮ ✇❤❡r❡ p = π∗(ϕ)✳ ❋♦r G✲✐♥✈❛r✐❛♥t s②♠♣❧❡❝t✐❝ ❢♦r♠ ωA = dγ = dΘ(A) ♦♥❡ ❤❛s t❤❡ G✲❡q✉✐✈❛r✐❛♥t ♠♦♠❡♥t✉♠ ♠❛♣ JA : T ∗P → T ∗

e G ❣✐✈❡♥ ❜②

JA = J0 ◦ A∗.❋♦r t❤❡ st❛♥❞❛r❞ s②♠♣❧❡❝t✐❝ ❢♦r♠ ω0 = dγ0 t❤❡ ♠♦♠❡♥t✉♠ ♠❛♣ ✐s J0(ϕ) = ϕ ◦ Tκπ∗(ϕ)(e). ■t ✐s r❡❛s♦♥❛❜❧❡ t♦ ❞❡✜♥❡ t❤❡ s♣❛❝❡ CanTGT ∗P := Θ(AutTGTP) ✇❤✐❝❤ ✐s ❛♥ AutTGTP✲✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡ ♦❢ t❤❡ s♣❛❝❡ CanT ∗P✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ✭✐✮ ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝❛♥♦♥✐❝❛❧ ❢♦r♠ Θ(A) ❜❡❧♦♥❣s t♦ CanTGT ∗P ✐❢ ❛♥❞ ♦♥❧② ✐❢ (Φ∗

g)∗Θ(A) = Θ(A)

❛♥❞ JA = J0 ◦ A∗ = J0✳ ✭✐✐✮ ❖♥❡ ❝❛♥ ❝♦♥s✐❞❡r ❛s t❤❡ ♦r❜✐t ♦❢ t❤❡ s✉❜❣r♦✉♣ t❛❦❡♥ t❤r♦✉❣❤ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❢r❡❡ ❛❝t✐♦♥ ❞❡✜♥❡❞ ✐♥ ✭✷✻✮✳ ✭✐✐✐✮ ■❢ ❛♥❞ t❤❡♥ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ✭✐✮ ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝❛♥♦♥✐❝❛❧ ❢♦r♠ Θ(A) ❜❡❧♦♥❣s t♦ CanTGT ∗P ✐❢ ❛♥❞ ♦♥❧② ✐❢ (Φ∗

g)∗Θ(A) = Θ(A)

❛♥❞ JA = J0 ◦ A∗ = J0✳ ✭✐✐✮ ❖♥❡ ❝❛♥ ❝♦♥s✐❞❡r CanTGT ∗P ❛s t❤❡ ♦r❜✐t ♦❢ t❤❡ s✉❜❣r♦✉♣ AutTGTP ⊂ Aut0TP t❛❦❡♥ t❤r♦✉❣❤ γ0 ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❢r❡❡ ❛❝t✐♦♥ L∗ ❞❡✜♥❡❞ ✐♥ ✭✷✻✮✳ ✭✐✐✐✮ ■❢ ❛♥❞ t❤❡♥ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

Pr♦♣♦s✐t✐♦♥ ✭✐✮ ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝❛♥♦♥✐❝❛❧ ❢♦r♠ Θ(A) ❜❡❧♦♥❣s t♦ CanTGT ∗P ✐❢ ❛♥❞ ♦♥❧② ✐❢ (Φ∗

g)∗Θ(A) = Θ(A)

❛♥❞ JA = J0 ◦ A∗ = J0✳ ✭✐✐✮ ❖♥❡ ❝❛♥ ❝♦♥s✐❞❡r CanTGT ∗P ❛s t❤❡ ♦r❜✐t ♦❢ t❤❡ s✉❜❣r♦✉♣ AutTGTP ⊂ Aut0TP t❛❦❡♥ t❤r♦✉❣❤ γ0 ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❢r❡❡ ❛❝t✐♦♥ L∗ ❞❡✜♥❡❞ ✐♥ ✭✷✻✮✳ ✭✐✐✐✮ ■❢ A ∈ Aut0TP ❛♥❞ L∗

A(CanTGT ∗P) ⊂ CanTGT ∗P t❤❡♥

A ∈ AutTGTP✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❈♦rr♦❧❛r② ❋✐①✐♥❣ ❛ ❝♦♥♥❡❝t✐♦♥ α ♦♥❡ ♦❜t❛✐♥s ❛♥ ❡♠❜❡❞❞✐♥❣ ια : ConnP(M, G) ֒ → CanTGT ∗P ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ια(α′) := ϕ◦Tπ∗(ϕ)+ϕ◦Tκπ∗(ϕ)(e)◦(α′

π∗(ϕ) −απ∗(ϕ))◦Tπ∗(ϕ).

✭✷✽✮

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

• ❆ G✲❡q✉✐✈❛r✐❛♥t ❞✐✛❡♦♠♦r♣❤✐s♠

T ∗P P × T ∗

e G

✲

Iα ❞❡♣❡♥❞❡♥t ♦♥ ❛ ✜①❡❞ ❝♦♥♥❡❝t✐♦♥ ✇❤❡r❡ ✐s t❤❡ t♦t❛❧ s♣❛❝❡ ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡ ❜❡✐♥❣ t❤❡ ♣✉❧❧❜❛❝❦ ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡ t♦ ❜② t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ♦♥ t❤❡ ❜❛s❡ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

• ❆ G✲❡q✉✐✈❛r✐❛♥t ❞✐✛❡♦♠♦r♣❤✐s♠

T ∗P P × T ∗

e G

✲

Iα ❞❡♣❡♥❞❡♥t ♦♥ ❛ ✜①❡❞ ❝♦♥♥❡❝t✐♦♥ α Iα(ϕ) := (Γ∗

α(π∗(ϕ))(ϕ), π∗(ϕ), ϕ ◦ Tκπ∗(ϕ))),

✇❤❡r❡ P := {( ˜ ϕ, p) ∈ T ∗M × P : ˜ π∗( ˜ ϕ) = µ(p)} ✐s t❤❡ t♦t❛❧ s♣❛❝❡ ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡ P(T ∗M, G) ❜❡✐♥❣ t❤❡ ♣✉❧❧❜❛❝❦ ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ❜✉♥❞❧❡ P(M, G) t♦ T ∗M ❜② t❤❡ ♣r♦❥❡❝t✐♦♥ ˜ π∗ : T ∗M → M ♦❢ T ∗M ♦♥ t❤❡ ❜❛s❡ M✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

T ∗P P × T ∗

e G

✲

Iα ✭✷✾✮ ✐s t❤❡ ✐♥✈❡rs❡ t♦ ✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❣r♦✉♣ ❛❝ts ♦♥ ✱ ✇❡ ❛❧s♦ ❝❛♥ ❞❡✜♥❡ t❤❡ ♥❛t✉r❛❧ r✐❣❤t ❛❝t✐♦♥ ♦❢ ♦♥ ❢♦r ✱ ❛♥❞ t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

T ∗P P × T ∗

e G

✲

Iα

✛

I−1

α

I−1

α ( ˜

ϕ, p, χ) = ˜ ϕ ◦ Tµ(p) + χ ◦ αp ✭✷✾✮ ✐s t❤❡ ✐♥✈❡rs❡ t♦ Iα✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❣r♦✉♣ AutTGTP ❛❝ts ♦♥ TP✱ ✇❡ ❛❧s♦ ❝❛♥ ❞❡✜♥❡ t❤❡ ♥❛t✉r❛❧ r✐❣❤t ❛❝t✐♦♥ ♦❢ AutTGTP ♦♥ T ∗P (A∗ϕ)(π∗(ϕ)) := ϕ ◦ A(π∗(ϕ)) ❢♦r A ∈ AutTGTP✱ ❛♥❞ t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ T ∗P Φ∗

g(ϕ)(pg) = (Tκg(p)−1)∗ϕ.

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❯s✐♥❣ Iα ✇❡ tr❛♥s♣♦rt ❛❜♦✈❡ ❛❝t✐♦♥s t♦ P × T ∗

e G✿

Λα(A)( ˜ ϕ, p, χ) := (Iα ◦ A∗ ◦ I−1

α )( ˜

ϕ, p, χ) = = (( ˜ ϕ ◦ Tµ(p) + χ ◦ αp) ◦ A(p) ◦ Γα(p), p, χ) ✭✸✵✮ ❛♥❞ ❜② ψ∗

g( ˜

ϕ, p, χ) := (Iα ◦ Φ∗

g ◦ I−1 α )( ˜

ϕ, p, χ) = ( ˜ ϕ, pg, Ad∗

g−1χ),

✭✸✶✮ r❡s♣❡❝t✐✈❡❧②✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❯s✐♥❣ I−1

α

: P × T ∗

e G → T ∗P ✇❡ ♣✉❧❧ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝❛♥♦♥✐❝❛❧

❢♦r♠ Θ(A) ❜❛❝❦ t♦ P × T ∗

e G✱ ✇❤❡r❡

Θ(A)ϕ, ξϕ := ϕ, A(π∗(ϕ))Tπ∗(ϕ))ξϕ, ✭✸✷✮ ❢♦r ξϕ ∈ Tϕ(T ∗P)✳ ❋♦r A = (✐❞TP + B)σα( ˜ A) ✇❡ ❤❛✈❡ (I−1

α )∗Θ(A)( ˜

ϕ, p, χ) = ✭✸✸✮ = ˜ ϕ◦ ˜ A(µ(p))◦T(˜ π∗◦pr1)( ˜ ϕ, p, χ)+χ◦αp◦A(p)◦Tpr2( ˜ ϕ, p, χ) = = pr∗

1(˜

Θ( ˜ A)( ˜ ϕ, p, χ) + pr3( ˜ ϕ, p, χ), pr∗

2(φA−1(α))( ˜

ϕ, p, χ), ✇❤❡r❡ pr3( ˜ ϕ, p, χ) := χ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❚❤❡ s②♠♣❧❡❝t✐❝ ❢♦r♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✭✸✸✮ ✐s ❣✐✈❡♥ ❜② d((I−1

α )∗Θ(A)) =

✭✸✹✮ = pr∗

1(d˜

Θ( ˜ A)) + d pr3

∧

, pr∗

2(φA−1(α)) + pr3, pr∗ 2(dφA−1(α)).

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❈♦♥s✐❞❡r✐♥❣ P ❛s t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡ ♦❢ ❛ ♣❤②s✐❝❛❧ s②st❡♠ ✇❤✐❝❤ ❤❛s ❛ s②♠♠❡tr② ❞❡s❝r✐❜❡❞ ❜② G ♦♥❡ ❝♦♥s❡q✉❡♥t❧② ❛ss✉♠❡s t❤❛t ✐ts ❍❛♠✐❧t♦♥✐❛♥ H ∈ C∞(T ∗P) ✐s ❛ G✲✐♥✈❛r✐❛♥t ❢✉♥❝t✐♦♥ ♦♥ T ∗P✱ ✐✳❡✳ H ◦ Φ∗

g = H ❢♦r g ∈ G✳ ❍❡♥❝❡ ✐t ✐s ♥❛t✉r❛❧ t♦ ❝♦♥s✐❞❡r

t❤❡ ❝❧❛ss ♦❢ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠s ♦♥ G✲s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ (T ∗P, ωA, J0) ✇✐t❤ ❛ G✲✐♥✈❛r✐❛♥t ❍❛♠✐❧t♦♥✐❛♥ H✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❯s✐♥❣ t❤❡ ✐s♦♠♦r♣❤✐s♠ (T ∗P, ωA, J0) ∼ = (P × T ∗

e G, (I−1 α )∗ωA, pr3)

♦❢ G✲s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ t❤❡ s②♠♣❧❡❝t✐❝ ❢♦r♠ (I−1

α )∗ωA

✇❛s ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛♥❞ t❤❡ ♠♦♠❡♥t✉♠ ♠❛♣ ✐s J0 ◦ Iα = pr3✱ ♦♥❡ ❞❡✜♥❡s t❤❡ G✲✐♥✈❛r✐❛♥t ❍❛♠✐❧t♦♥✐❛♥ H ∈ C∞(P × T ∗

e G) ❛s ❢♦❧❧♦✇s

H( ˜ ϕ, p, χ) := ( ˜ H ◦ µ)( ˜ ϕ, p, χ) + (C ◦ pr3)( ˜ ϕ, p, χ), ✇❤❡r❡ µ : P → T ∗M ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ t♦t❛❧ s♣❛❝❡ P ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ G✲❜✉♥❞❧❡ P(T ∗M, G) ♦♥ t❤❡ ❜❛s❡ T ∗M✱ ˜ H ∈ C∞(T ∗M) ❛♥❞ C ∈ C∞(T ∗

e G) ✐s ❈❛s✐♠✐r ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

st❛♥❞❛r❞ ▲✐❡ P♦✐ss♦♥ str✉❝t✉r❡ ♦❢ T ∗

e G✳

❈♦♠✐♥❣ ❜❛❝❦ t♦ t❤❡ ♣❤❛s❡ s♣❛❝❡ ♦♥❡ ♦❜t❛✐♥s t❤❡ ✲❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ✇✐t❤ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ▲❡t ✉s str❡ss t❤❛t ✐♥ t❤❡ ❝❛s❡ ♦❢ ♦♥❧② t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ t❤❡ s②st❡♠ ❞❡♣❡♥❞s ♦♥ ❛♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♦♥❧② s②♠♣❧❡❝t✐❝ ❢♦r♠ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❯s✐♥❣ t❤❡ ✐s♦♠♦r♣❤✐s♠ (T ∗P, ωA, J0) ∼ = (P × T ∗

e G, (I−1 α )∗ωA, pr3)

♦❢ G✲s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ t❤❡ s②♠♣❧❡❝t✐❝ ❢♦r♠ (I−1

α )∗ωA

✇❛s ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛♥❞ t❤❡ ♠♦♠❡♥t✉♠ ♠❛♣ ✐s J0 ◦ Iα = pr3✱ ♦♥❡ ❞❡✜♥❡s t❤❡ G✲✐♥✈❛r✐❛♥t ❍❛♠✐❧t♦♥✐❛♥ H ∈ C∞(P × T ∗

e G) ❛s ❢♦❧❧♦✇s

H( ˜ ϕ, p, χ) := ( ˜ H ◦ µ)( ˜ ϕ, p, χ) + (C ◦ pr3)( ˜ ϕ, p, χ), ✇❤❡r❡ µ : P → T ∗M ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ t♦t❛❧ s♣❛❝❡ P ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ G✲❜✉♥❞❧❡ P(T ∗M, G) ♦♥ t❤❡ ❜❛s❡ T ∗M✱ ˜ H ∈ C∞(T ∗M) ❛♥❞ C ∈ C∞(T ∗

e G) ✐s ❈❛s✐♠✐r ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

st❛♥❞❛r❞ ▲✐❡ P♦✐ss♦♥ str✉❝t✉r❡ ♦❢ T ∗

e G✳ ❈♦♠✐♥❣ ❜❛❝❦ t♦ t❤❡ ♣❤❛s❡

s♣❛❝❡ (T ∗P, ωA, J0) ♦♥❡ ♦❜t❛✐♥s t❤❡ G✲❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ✇✐t❤ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ Hα(ϕ) := (H ◦ Iα)(ϕ) = ( ˜ H ◦ Γ∗

α)(ϕ) + (C ◦ J0)(ϕ).

▲❡t ✉s str❡ss t❤❛t ✐♥ t❤❡ ❝❛s❡ ♦❢ ♦♥❧② t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ t❤❡ s②st❡♠ ❞❡♣❡♥❞s ♦♥ ❛♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♦♥❧② s②♠♣❧❡❝t✐❝ ❢♦r♠ ✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❯s✐♥❣ t❤❡ ✐s♦♠♦r♣❤✐s♠ (T ∗P, ωA, J0) ∼ = (P × T ∗

e G, (I−1 α )∗ωA, pr3)

♦❢ G✲s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s✱ ✇❤❡r❡ t❤❡ s②♠♣❧❡❝t✐❝ ❢♦r♠ (I−1

α )∗ωA

✇❛s ♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛♥❞ t❤❡ ♠♦♠❡♥t✉♠ ♠❛♣ ✐s J0 ◦ Iα = pr3✱ ♦♥❡ ❞❡✜♥❡s t❤❡ G✲✐♥✈❛r✐❛♥t ❍❛♠✐❧t♦♥✐❛♥ H ∈ C∞(P × T ∗

e G) ❛s ❢♦❧❧♦✇s

H( ˜ ϕ, p, χ) := ( ˜ H ◦ µ)( ˜ ϕ, p, χ) + (C ◦ pr3)( ˜ ϕ, p, χ), ✇❤❡r❡ µ : P → T ∗M ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ t♦t❛❧ s♣❛❝❡ P ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ G✲❜✉♥❞❧❡ P(T ∗M, G) ♦♥ t❤❡ ❜❛s❡ T ∗M✱ ˜ H ∈ C∞(T ∗M) ❛♥❞ C ∈ C∞(T ∗

e G) ✐s ❈❛s✐♠✐r ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

st❛♥❞❛r❞ ▲✐❡ P♦✐ss♦♥ str✉❝t✉r❡ ♦❢ T ∗

e G✳ ❈♦♠✐♥❣ ❜❛❝❦ t♦ t❤❡ ♣❤❛s❡

s♣❛❝❡ (T ∗P, ωA, J0) ♦♥❡ ♦❜t❛✐♥s t❤❡ G✲❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ✇✐t❤ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ Hα(ϕ) := (H ◦ Iα)(ϕ) = ( ˜ H ◦ Γ∗

α)(ϕ) + (C ◦ J0)(ϕ).

▲❡t ✉s str❡ss t❤❛t ✐♥ t❤❡ ❝❛s❡ ♦❢ (T ∗P, ωA, J0, Hα) ♦♥❧② t❤❡ ❍❛♠✐❧t♦♥✐❛♥ Hα ♦❢ t❤❡ s②st❡♠ ❞❡♣❡♥❞s ♦♥ α ∈ ConnP(M, G) ❛♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ (P × T ∗

e G, (I−1 α )∗ωA, pr3, H) t❤❡ ♦♥❧② s②♠♣❧❡❝t✐❝

❢♦r♠ (I−1

α )∗ωA✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❘❊❋❊❘❊◆❈❊❙✿

✶ ❏✳ P✳ ❉✉❢♦✉r✱ ◆✳ ❚✳❩✉♥❣✳ P♦✐ss♦♥ ❙tr✉❝t✉r❡s ❛♥❞ t❤❡✐r ♥♦r♠❛❧

❢♦r♠s✱ ✷✹✷✱ ❇✐r❦❤❛✉s❡r ❱❡r❧❛❣✱ ✷✵✵✵✳

✷ ❑✳ ❈✳ ❍✳ ▼❛❝❦❡♥③✐❡✳●❡♥❡r❛❧ t❤❡♦r② ♦❢ ▲✐❡ ❣r♦✉♣♦✐❞s ❛♥❞ ▲✐❡

❛❧❣❡❜r♦✐❞s✱ ✷✶✸✱ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ▲❡❝t✉r❡ ◆♦t❡ ❙❡r✐❡s✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✷✵✵✺✳

✸ ❑✳ ❈✳ ❍✳ ▼❛❝❦❡♥③✐❡✱ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ❆✳❙❧✐➺❡✇s❦❛✳ P♦✐ss♦♥

❣❡♦♠❡tr② r❡❧❛t❡❞ t♦ ❆t✐②❛❤ s❡q✉❡♥❝❡s✱ ❙■●▼❆ ✶✹ ✭✷✵✶✽✮✱ ✵✵✺

✹ ❙✳❙t❡r♥❜❡r❣✳ ▼✐♥✐♠❛❧ ❝♦✉♣❧✐♥❣ ❛♥❞ t❤❡ s②♠♣❧❡❝t✐❝ ♠❡❝❤❛♥✐❝s

♦❢ ❛ ❝❧❛ss✐❝❛❧ ♣❛rt✐❝❧❡ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❨❛♥❣✲▼✐❧❧s ✜❡❧❞✳ Pr♦❝✳◆❛t❧✳❆❝❛❞✳❙❝✐✳❯❙❆✳✱✼✹✭✶✷✮✿✺✷✺✸✲✺✷✺✹✱ ✶✾✽✽✳

✺ ❆✳ ❲❡✐♥st❡✐♥✳❆ ✉♥✐✈❡rs❛❧ ♣❤❛s❡ s♣❛❝❡ ❢♦r ♣❛rt✐❝❧❡s ✐♥

❨❛♥❣✲▼✐❧❧s ✜❡❧❞s✳ ▲❡tt✳ ▼❛t❤✳ P❤②s✳✱ ✷✭✺✮✿✹✶✼✕✹✷✵✱ ✶✾✼✼✴✼✽✳

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s

❚❍❆◆❑ ❨❖❯ ❋❖❘ ❆❚❚❊◆❚■❖◆

❙②♠♠❡tr✐❡s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ❝♦♥♥❡❝t✐♦♥s ♦♥ ❛ ♣r✐♥❝✐♣❛❧ ●✲❜✉♥❞❧❡ ❛♥❞ r❡❧❛t❡❞ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡s