2021 - 9- 27 Kai her geometry

Thf ①Rijni = - s 'J÷¥÷2 2-

i2 zi

iii.e- = a÷X÷°÷ - ?÷#÷¥÷- )=

= g( d-nie . ?÷(FÉ ¥ )) = F-E) snap a)- EEE

iii. = IIe = 9"¥÷- =sni.IE?;-..-;-:Rijri--9nj?-zi)

#= is; ,ñ -99 Pig ÷¥If szi

=É2 zi , zi

- g" 5 -29k,-

y zi

¥27T #

Majority of Kinter geometersuse Rijn e- = -

¥ + .. . .

- (2)2 tis Ii

Met (M , g) Riemannian mfd . ②el

,- > en orthonormal frane . of TM .

F-Ric Cxi) = É RCX

,ei,Y, ei )

i=l# '

Ric CY,✗7=21%144)

pie is symmetnithe = Rij dxi@ dxi Rij =Rjithe is called the Ricci curvature

or Ricci tensor .

Rij = gkl Rinji.

-

= Rin;"=

(Mcg ) Kii when mfd .

Rijkn 1- Rik ni + R = °

.

" pie - ise)- Ri

"

ik -

③: . Rijkk = Rikin = Rij€¥raTm for kéwlu

Majority of Kincer genets use with 4)

Rij = Rijkk The same as

z#-

oars.

Prop_ Rij = -¥ñ long detcske )i ^If

# In general for invertible matrixdldet A) =dd-A.to#dA7---

EIÉ=Soif!j = g-

i I 2%ñ_-0Zi__

= ÷. log dethrone ).

ID:- Rijhn = - ghi Rijn e- = -gkiÉ. Snag

zzi-

.

= - ÷..

Pile = - ÷.log detg!

④ = JI = ( Rijn e- dzknd -2T )•

dot one is#-

curvature form ( 2-foul

④

matrix.

Det dit ( I + t É④) m=di①H .

I ⇐

= I + te,+ t'

ce t - - + tmcm

Fact cp (g) is a real closed zp- form .

de Khan class [epcq )] is indop off .We do . it only for p= 1

.( later )

Det [ cpcgid-itq.CM ) c- It ,¥ (M )is called the p - th Chern class.

⑤

Look at ite case p= 1.

4197 = ¥ to④ = =É' Riinidthndit-

Ra e-"

i = Rai

=- É si log det 19 ;-)

If 9'is another Kinter form , then

- Fast log det 19 :;-) - l - ÉIoo logdetls!;D=-

• say-Éidet 19 'i ;-)

Lemma-H

c- ICM),

i. e. indep ofdet 19

'

;-)local coordinate

.

proofIf 2-1

, :-, -2"

is Lto cord onU

.

①

w'

,- ,w" i halo cmd on

✓

U Tv.v -1-4

glivi.¥1 -71¥"÷ . )2

i. detl91-wi.IE;))=|det¥detfgl-z.ro?-i- 1)

In the same way

detloitwi,

1) =/ det¥detst*i*¥Yi.÷i)=÷¥ndlt ( KC ,÷.

. ;É.)) out 151¥ , ¥-1 )

2+5

Thus ③city - alk)=Ñ - Flog -4 )

detg'

'

- 41913=419's]Lennie : If ✗ is a real exact cp ,p ) - formthen 7- real lp-1 , p- 1) ifnup sit .

4- is JP .

In particular for p=l , i. -e .

fn exact 11 > 1) ✗, we

have 4,0 ) - form

i. e. a smooth function 9 e- c- (M)

such that ✗ = i 254.

( This is called the 25 - lemma . )

This is an application of Hodge Theory.

see.

Aitutaki LectureHotespri-ngh.siKobayashi : Transformationgroups

in differential geometryspringer . A#i . #