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a = :X"

vis the musial Bomwphism b : Hm) → rtm ) 14 muse * : sum) → Hm) )The cod thy ht Riemannian 3- fb B Hit we here a cross product Are a veetr triple paoht

[ X , Y ,⇒ ;gp1 X, Yx⇒ = Att [

xpYpEDAs a result, he also he an domfhs , *↳ : Hm ) film ) gun by* KDPIY, A = [ X , 4, Dpwhen X , 4, Z are with in arthonoml lw.at . g) lord cows at p .This is def smrlh & multitier, as well as a Hmtysne * KDPIY, Y) = [ X , 4 , 'D Iout [ Xp Yp Yp ) = 0

We already swtht Nl m ) ± Calm) , so we set th flbwy conmtlm diym :

sum ) d- sun ) dosilm ) d- Mm)

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fray he whvh exntf specializes to gm th dwveu & grokt may mfd & , adding , the out m 3- nfds .

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Speed cases f the Same thuan : Stokes ' The oenfr diftehl fens .

2commits : 0 In the diagram,

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you stt in sum ) w si 4 Lr, ftt nth , silm) r Alm ) , sue the hghrd's are all Inuit)

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Then, as wolf chin aphxs , u the the homology , getty the kth de khan cohomology grip

Hhaalm) : = twoin dktl

𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝2𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝 silml ± Xtmhsilml R silml = Came sik ) .

In get,if M is an nnaitbl, then silm ) ± th lm) F k .

This will the specialize to a resin f panaidnlilg fr th dekhan cohomology gps .

=so Algebra & ExteriorAth

-

To define differentit fans pgaf ( & to tmtms into rgndedafebm ),

n need to do some Hired akin .

Gm veetr spry V & W, define the thsw podnt VOW to be the met spa f dents now fr u EV, we W, st .𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝1𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝 ( v , tud Q W = v , ow + how

𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝2𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝 V @ ( w , tw 2) = VQW , t VQ wz

𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝3𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀼𐀽𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝 a ( vow ) = ( a DOW = v@ law)

he '. Tf V = IRM, W= Rn

,then Job corps to the mxnmtrix Tut = [jD[ w , . . . w . )

Them (Animal Pgnf) . . If 4 : Vxw - Vow is the np gun by lv, who vow , & Fi Vxw → U is kliw , th F '. law np

I : V@wosnmehytrdhsmcanmti.V xw

F-U

1inati'VQW

Mruv,this way duties VOW : ay us . satsty this party is Bomphz to Vxow .

The point B tht thsw prodwb are a wg of twny biker ( or, me gulf , multi hw) maps into thumps .

Pq : If e , , . , , em is . bn 's tr V R fi , - ,th is a brisk W, then { eixofj } i., is a bad fr how .

In Patil, din Vow = mn .

Df : The for spare of type U.s ) asoitsl 4 V B th rear spa

Vs = Vxo . . . QVQ V*@ . ..*÷✓ s

he '. An in w psdnt ( i ) on V is a symmetr 6,2 )- tensor on V

'

.

L., 7 detb a bilker hp Vxv → R . By the

universal pgef , th . ] idms a ! liner np V@V→R .

so L ' ; ) unruly corresponds to an

demtf1VxvY-Vt@vtnTheAhutSumT1H-QoVrsuheVo0-gmlfeldfVisthetnswalsebnfV.T

( V) B nncomnktne ( Yoh F KDV , ) , associate & ( bit graded .

Nw, define the tier

bundle of type Iris) nr a nfd M to be

TIM ) = Yen ltpmlr

whih B a swath Nfl.A for field of type ks ) mm is a susth seetm f the Irs) tmubkk ; tit is, a

smith mp M → Til M)

hf : ' A vectr field an M is a 11,01 - tensor fed m M .i A Riemannian mdnz an M B a symwfnz 10,2)

- tenser field

. A 1- form B a ( 0,1 ) - tar field .