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Page 1: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

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Page 2: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

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Page 3: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

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d�{bi�d'NuR�V�d�QC^'J�LPJ�J�H�S'abfxT'NPQhjc3¸'d~¸P»�à ¸�`�e�^'J]\|NPVPR�{�J|{�VON�v�Q�abR�S'T'�

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n2 j�¥�V�i J�{�T�N3�4.5 X 6= 0

fxa l a�NuR�{�`'S'J�VPR�S'a�^'VPR�{�N�{�VPNuv¦Q�abR�S'T'�¦J�Hbi/abVPR�c�j(��a�W~W�a�W_NPVui T_J�`'S'N�rl Nui T�R�Nui�c�^'NPW�J�R�J�S�J�KqQ�abR�S�T'�'g

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J�H�S�abfOJ�Qhn `'Jbi ^�a�d1i T�^'NOK'^'abdMLOS�y�`�`�a�d'{bi'd'NPR�V�dMJ�Hbi a�VPR�c�F {R

n2 n�a�Vui N l J�{a�rR�Nui�c�^'J�T�QC^'J�LPJ�J�H�S'abfxT'NPQhj

��Gi�d l J�W�a�f]abR�Nui c�VuR�{a¡R�J�LPJ'nhU�R�J�R�J�T�i�T T'^�J�N[`'J l QC^'J]\|NOVuR�{�J�NO{�W�i T l J�r{ae`'S'J�VuR�S�ab^'VPR�{a1d�{bi�d'NuR�V�d�QC^'J�LPJ�J�H�S'abfxT'NPQ U'a�VPR�J1T�VO`'Jbi�c�fOy�FmRg&~¸OÅ¢¹/¸ä¶[h@Å» ¸�$�#b» Å%ikj�h»�Âal�Ã�Ã�j

m�a�VOVPQCJ�R�S'T�Q {¦`'S'J�VPR�S'a�^'VPR�{�NRn

V�W�J�J�S l T�^�abR�a�QCTx1, . . . , xn

QC^�J�\pNOVPR�{�JX ⊂ Rn

n�VOJ�VPR�J]d�opNON%T�f�R�J�U�NOW-n�W�J�J�S l T'^�abR�g W�J�R�J�S'g(v�y l J�{bi NuR�{�J�Sd�FmR~VOT�VPR�N�rQCN�y�S�ab{�^'NO^�T'K

f1(x1, . . . , xn) = 0,

· · ·fs(x

1, . . . , xn) = 0.e�abR�S�T'��a

Jx =(

∂fi

∂xj

x

)

∈ M(s, n)^�abfxgh{�a�NPR�V�d�¶�ÀA&I¹ Ãnl ¸.ipoGÂ�Å]Æ�Ã�suR�J�K@VPT�r

VPR�NPQCg%n�a|NPNGJ�`'S'N l Nui T�R�Nui cq� NOV�i T�J�^�J�`'S'N l Nui NP^ �q$�Â�Å]Æ�Ã�À�» ÅÚ¶%jr °�s�¯ °�t�u(µ }CVui T~S�ab^'LwvOx

Jx = k{�VOF l yp^�a

XnbR�J%QC^�J�\pNOVPR�{�J

Xd'{�i�d'NuR�V�d

L¢i/a l W'T�Q"QC^'J�LPJ�J�H�S�a�fxT'NPQ�S�abfOQCNOS'^�J�VPR�T¡�n− k

�ujy�a�WXVPU'T�R�abR�c�S�a�^�LqQ�abR�S'T��'g{z%W�J�H�T�|}� VPT'VPR�NOQCJ�K9y�S�a�{�^�NO^'T'K VO{bd�f]ab^'J

J�R�J�H�S�ab\|NP^'T'NF : Rn(x1, . . . , xn) → Rs(y1, . . . , ys)

F (x1, . . . , xn) = (y1, . . . , ys),

yi = fi(x1, . . . , xn)

X = F−1(0)��i'd�i FmH�J�K�R�J�U'W'T

x0 ∈ RnJ�`'S�N l N�i NO^ l T'����NOS�NO^'�'T'a�i

dx0FJ�R�J�H�S�ab\|NP^'T�d

F{IsPR�J�K�R�J�U'W�N�Ëdx0

F : Rn → Rsn�W�JbR�J�S'ghK�{�NOW�R�J�S

a = (a1, . . . , an)JbR�J�H�S'ab\~a�NuR

{_{�NPW�R�J�Sb = (b1, . . . , bs)

`�J¦��J�S�Q�y�i�N

bi = (dx0F (a))i =

n∑

j=1

∂fi

∂xj

x0

ai = (Jx0a)i

£:R�J'n]J�U�NO{�T l ^'J'nPi T�^'NOK'^�J�N�J�R�J�H�S�ab\|NP^'T'N�jx¥mR�{�NPS\ l abNPR�V�d�n�U�R�J)vOxJx0

= 3�~�������3 x0F� sPR�J�{�g6R�NOW�a�NuR�T�fGVOJ�J�R�{�NuR�VuR�{�y�Fmo�T�vqJ�`'S'N l Nui NP^'T'K ��j

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Page 4: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

�%S'T'{�N l NPQ y l J�H�^'ghK¡VO`'J�VOJ�Hq{�ghU�T'Vui NP^'T�d l T'����NOS�NO^'�'T'a�i/a�� suR�J�R[`'S�T'NOQU�a�VPR�J~T'VO`'J�i c�fOy�NPR�V�deT�{ l S�y�LPT�v�f]a l a]U�a�v'�ujn��R�J�H�g {�ghU'T'V�i T�R�c

dx0F (a)

n�S�a�V�rVOQCJ�R�S'T'Q9W'S'T'{by�F

γ(t){

Rnn `'S�J]v�J l d�o�y�F�U'NOS'Nuf

x0

T�T'QCNOFto�y�FÊ{x0

W�a�Vxa�rR�Nui�c�^'ghKe{�NPW�R�J�S

a� ^�a�S�T'VPy�K�R�N�S'T'Vuy�^'J�W-ÏÐ�uj���J|NPVPR�c

γ(0) = x0,d

dt

t=0

γ(t) = a

��J�L l a���R�Nuv�^'T'U�NOVOW�abd�`'S'J�{�NOS'W�a�

dx0F (a) =

d

dt

t=0

F (γ(t))

\ ÀbÁ�ÀA]�À��(`C��J�W�abfxabR�c'nU�R�J�LPS�y�`'`�aO(n,R)

d�{bi�d'NuR�V�d_L¢i/a l W�T'Q@QC^'J�LPJ�J�H�S'a�rfxT'NPQ"T�^�a�K�R�T�NON�S�a�fxQCNOS�^'J�VPR�c'j

c3¸'d~¸P»�à ¸�`��=S�y�`'`�aO(n,R)

{bi J]\|NP^�a¦{_`'S'J�VuR�S�ab^'VPR�{�JRn2 j��(J�J�R�^�J�wpNP^'T'N

XXT = 1l abNPR

n2 y�S'a�{�^'NP^'T'K-jC�%R�J�H�g l J�W�a�f]abR�c'n�U�R�J�LOS�y�`�`�a�J�S�R�J�LOJ�^'a�i c�^'g(vQ�abR�S'T���d'{bi'd'NPR�V�dMQC^'J�LPJ�J�H�S'abfxT'NPQ `'J l VPU'T�R�a�NPQ�S'a�^'L�Q�a�R�S'T'��g�z�W�J�H�T�sPR�J�KVOT'VuR�NOQCg y�S'a�{�^'NP^'T'K-j��Gi�d[suR�J�LPJ'n=VOJ�L¢i a�VO^'J�J�`�T'Vxa�^�^'J�K[{�g6wpN~W�J�^'VuR�S�y�W'��T'T-nJ�`'S'N l Nui T�Q�JbR�J�H�S'ab\|NO^�T'N

F : Rn2 → Rn2 `'J���J�S'Q�y]i NF (X) = XXT − 1

jz�VP^'J'n'U�R�J

O(n,R) = F−1(0)jb1(ghU'T'V�i T'Q l T'���GNPS'NO^��'T�a�i

dX0F : Rn2 → Rn2

{�R�J�U'W�NX0 ∈ O(n,R)

j���i'dqsPR�J�LOJ�S�a�VPVOQCJ�R�S'T'Q W'S'T'{by�Fγ(t)

{Rn2 n�R�a�W�y�F�n

U�R�Jγ(0) = X0

T ddt

t=0γ(t) = X

r3^'NOW�J�R�J�S�abd�Q�abR�S�T'��a�jn��J�L l a|T'QCNONPQ

A := dX0F (X) =

d

dt

t=0

F (γ(t)) =d

dt

t=0

[γ(t)γ(t)T − 1] = XXT0 +X0X

T

k U�NO{�T l ^'J'n U�R�JAr3VOT'QCQCNuR�S'T�U'NOVPW�a�dqQ�abR�S�T'��a�j/£:R�J�J�fO^�a]U�a�NPR�n U�R�J ����3 X0

F`'S'T'^�a l i NP\_T�R9QC^'J]\|NPVPR�{�y�VOT�QCQCNPR�S�T'U'NOVPW'T�v�Q�abR�S'T'�-j[� l S�y�LOJ�K VPR�J�S'J�^�g%nl i�d�i�FIH�J�K�VOT'QCQCNuR�S'T'U�NOVOW�J�K Q�abR�S'T'�'g

An3QCJ]\�^'Jz^�a�K�R�TXQ�abR�S�T'��y

Xn�R�a�r

W�y�F�n�U�R�JA = XXT

0 + X0XTnCL l N

X0 ∈ O(n,R)j��Gi�d�sPR�J�LOJ l J�VPR�a�R�J�U'^�J

`'Jbi J]\_T�R�cX = (1/2)AX0

j��a�W'T'Q�J�H�S�abfxJ�Qhn�J�H�S�abf ����3 X0

FJ�R�J�H�S�ab\|NP^'T�d

dX0F

VOJ�{�`'a l a�NuR[V�`'S'JbrVPR�S'a�^'VPR�{�J�QzVOT'QCQCNuR�S'T'U�NOVOW�T�v_Q�abR�S'T��-j��3i N l J�{abR�Nui c�^'J'n�S�ab^'L6Q�abR�S'T'�'g�z�W�JbrH�T�VOT'VuR�NOQCg%n J�`�T'VOgh{abFmopNPK�LPS�y�`'`�y

O(n,R)n�`�J�VPR�J�d�^'NO^�n'R�j�N

O(n,R)rCL¢i/a l r

W�J�NGQC^�J�LOJ�J�H�S�abfOT'N�j1(ghU�T'Vui T�Q"S�abfOQCNOS'^'J�VPR�c_sPR�J�LOJ¦QC^'J�LPJ�J�H�S�a�fxT�d�j�%QCNONPQ�vOx

JX0= 3�~�������3 X0

F = 3�~�� { VOT�QCQCNPR�S�T'U'NOVPW'T'NpQ�a�R�S'T'��g } =(1/2)n(n+ 1)

j��%J�sPR�J�Q�y 3�~�� O(n,R) = n2 − (1/2)n(n+ 1) = (1/2)n(n− 1)j

Page 5: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

1¡R�a�W�J�QM\|N l yv�N�QCJ]\�^'J l J�W�abfxabR�c'n�U�R�J�{�VON�`�S'T'{�N l NO^'^�ghN%{�g6wpNmLOS�y�`'`'gd'{bi�d�FmR�V�d�QC^'J�LOJ�J�H�S�abfxT�d'QCT-j+�

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{¦N l T'^'T'�'Nbjc3¸'d~¸P»�à ¸�`t��y�VuR�c

X(t)L¢i/a l W�a�d W'S�T'{abd�nhi�NP\~abopabd {�LPS�y�`'`'N

SL(n,R)T `�S'JxvJ l d�opabd U'NOS�NPfzN l T'^�T'��y/ËX(0) = 1

jw��J�L l a@W�a�VxabR�Nui c�^'ghK {�NPW�R�J�S WsPR�J�K@W'S'T'{�J�K@{�N l T'^'T��'N�`'S�T'^�a l i�NP\_T�RqW�a�Vxa�R�Nui�c�^'J�Q�y1`'S'J�VPR�S'a�^'VPR�{�y/Ë

A :=ddt

t=0X(t) ∈ TeSL(n,R)

jC��a�WpW�a�W|W'S�T'{abd�i�NP\_T�R%{�LPS�y�`'`'Nhy�^'T'QCJ l y]i�d�S'^'g(vQ�abR�S'T��-n�R�J 3�4.5 X(t) = 1

j:��T'����NOS�NO^'�'T�S�y�d�sPR�J_S�ab{�NO^'VuR�{�J_`'Jbi�y�U'T�QhË

0 =d

dt

t=0

3�4'5 X(t) = � v ddt

t=0

X(t) = � v A��a�W'T�Q J�H�S�abfOJ�Q�^�abw�NeW�a�VxabR�Nui c�^'J�N[`'S'J�VuR�S�ab^'VPR�{�J¡i�NP\_T�RM{M`'S'J�VPR�S'a�^'VPR�{�NQ�abR�S'T��qV�^�y]i NP{�ghQXVui N l J�Qhj���J�W�ab\|NOQhn/U�R�J�J�^'J_V�^'T'Q�VOJ�{�`'a l a�NuR�j��Gi�d�sPR�J�rLOJ@^'NPJ�HxvJ l T'QCJ l i�d"`'S�J�T�fx{�Jbi c�^'J�K"Q�abR�S�T'�'g

AR�a�W�J�K-n:U�R�J � v A = 0

`�JbrVPR�S�J�T�R�c|W�S'T'{�y�F

X(t){GLOS�y�`'`'N

SL(n,R)n�`�S'JxvJ l d�o�y�F U'NOS�NPf�N l T'^'T���y~R�a�W-n

U�R�J�H�gA

H�g(i¡W�a�VOabR�N�i c�^'ghQ�{�NOW�R�J�S'J�Q�W�sPR�J�KMW'S'T'{�J�K@{�N l T'^'T'��N�j�1�W�a]U'N�rVPR�{�N~R�abW�J�K1W�S'T'{�J�K[{�Jbfxc�QCNPQ

X(t) = exp (tA)jn��NOK'VuR�{�T�R�Nui c�^'J'n

X(0) = 1T

ddt

t=0X(t) = A

j k VuR�a�NPR�V�d_`�J�W�abf]abR�c'n�U�R�J%W'S'T'{�abd�i Nu\�T�R%{�LOS�y�`�`'N�j�1(J�VO`'Jbi�cbrfOy�NPQCV�deT�fO{�NOVuR�^'ghQXT�f�i�T'^'NOK�^'J�K�a�i LONPH�S'g S'a�{�NO^�VPR�{�J�Q

3b4.5 exp (tA) = exp � v (tA)

k R�VOF l a¦Vui N l y�NuR�n�U�R�J 3b4.5 X(t) = 1j

��R�a�W-nOW�abVxabR�Nui c�^�J�N:`�S'J�VPR�S�a�^'VuR�{�JtW�LOS�y�`'`�N3y�^'T'QCJ l y]i�d�S'^'g(vGQ�abR�S'T'��VOJ�{br`�a l a�NuR%V(`�S'J�VPR�S�a�^'VuR�{�J�QzQ�abR�S'T'�|V:^�y]i NO{�ghQ[V�i N l J�Qhj

�\ ÀbÁ�ÀA]�À���`J�pJ�W�a�f]abR�c'n:U�R�J@LPS�y�`'`�aSU(2,R)

LOJ�QCNOJ�QCJ�S��G^'azR�S'N�v�QCNOS'^'J�KVO��NOS'N

S3 jc3¸'d~¸P»�à ¸�`=�m^�a�i T�f_y�Vui J�{�T'K-nZW�J�R�J�S�ghQ l Jbi�\_^�a�y l J�{bi�NPR�{�J�Sd�R�c�Q�abR�S�T'��aT�fSU(2,R)

`'J�W�a�fxgh{a�NuR�n�U�R�J�`'S'J�T�fx{�J�i c�^'ghKps�i NOQCNP^�R%LPS�y�`'`'gMf]a�`�T'VOgh{abNPR�V�d{I{�T l N (

z w−w z

) n�L l Nz, w ∈ C

R�a�W'T�N�n�U�R�J |z|2+|w|2 = 1j*��JtNOVuR�c

(z, w) ∈S3 ⊂ C

2 j�

� ?CÙ�íäÙPå'ÔPåMèbÝ·ê�ÝPÞ-èbÕPÝpíäØ�Ö�ìxÛ�Ø�ìxÖ�áCÞ ÛOÔPØ�ÔuÖ�áZå'â�ÔPó�êbÙÚè�Ù�é�ئÔPà�â�íäÙuï�ï�á=Ýpà�Ö�ÔOí·Ø�ÖbÙ�ï�íäØ�ÕPÙ*�íäØ�Ö�ìxÛ�Ø�ìxÖ�Ù�ãäÖ�ìxà�à�á â í·Ø�Ö�ìOÛ�Ø�ìxÖ�Ù¡ãëêbÙÚèbÛxÔuãÈÔ¡å'ï�ÔuãÈÔuÔuóPÖ�Ù�çÈâ]×�×�Õ�ê�×�é�Øuí«×9í·Ôuãëê�Ùuí·ÔPÕPÙuï�ï�á-îå'â�ð���ï�â í·ÔPãëêbÙuíäÔuÕPÙuï�á Õ�Ø�ÔPå�íÈå'áZíºê�ÝPÞmß�ØuÔ�ãäÖ�ìOà�à�ÔuÕPá=Ý[Ôuà�ÝÈÖ�Ù�=�â�â ÿ à�Ö�Ôuâ�çÈÕuÝ·èbÝäï�â�Ý[âÕPç·×�Ø�â�Ý~ÔuóPÖ�Ù�Ø�ï�ÔuãäÔ�ô·êbÝÈå'Ýäï�Ø�Ù��I×�Õ�ê�×�é Øuí«×eãëêbÙ�è�Û�â�å'â ÿ Õ�íÈå'áZíºê�Ý~íäØ�Ö�ìxÛ�Ø�ìxÖ�á å'ï�ÔPãäÔPÔuóuÖbÙÚîçÈâ]×A�ÎÔPØ�ÔuóPÖ�ÙÚÜ(Ýäï�â�×�å'â�ð���ÙuÛ�â�Ý�à�Ö�ÔPí·ØuÖ�Ù�ï�íäØ�ÕPÙ(ï�Ùuçäá=ÕOÙ�é ØPí«×�E��CT.�*�A�Èù�I[��I�ð�

Page 6: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

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F (X) = XXT − 1`'J l VOU�T�R�a�^'^'J�LOJ�{_i FmH�J�K

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XT = X�ujC�%J�suR�J�Q�y¡LPS�y�`'`�a1y�^'T�R�a�S'^'g(v�Q�abR�S�T'��H�y l NuR�L¢i a l W'T'Q

QC^'J�LOJ�J�H�S�abfOT'NOQ�S�abfOQCNOS'^�J�VPR�T 3�~�� U(n) = 2n2− 3�~�� { sOS�QCT�R�J�{�g Q�abR�S�T'�'g } =n2 j

© jx��J l S�J�H�^'J�NCJ�`'T'Vxab^'T'N3W�i/a�VPVOT'U'NPVOW'T�v�Q�a�R�S'T'U�^'g(v�LOS�y�`'`GQCJ]\�^'J ^�abK�R�TG{W'^'T�\_W�N:��j��:j��pJ�QCNP^'W�Jª �"�Ã�¶[«�¿�¸uÂa&|Ãn]�¸.¬PÂÀ%$¦½�¸OÅÚ¶�¸.&I¹ Ã($/ ºn*��i/ab{a%§�n § §�j�¬%T�\pN`'S'T'{�N l NO^'a¦R�a�Hbi�T'��a~J�R�{�NuR�J�{

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r rn2

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SO(n,R)­ ­

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MnnNm

r l {a¡L¢i/a l W'T�v QC^'J�LPJ�J�H�S'abfxT�d�j)m�a�VPVOQ�abR�S'T'{a�dMn

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S�ab\|NP^'T�d�vf : Mn → Nm

j·��i/a l W'T'N�VuR�S�y�W�Rby�S'g%nCf]a l ab^'^'ghN�^�a�QC^'J�LOJ�J�H�S�a�rfxT�d�v�n�`'Jbfx{�Jbi'd'FmRz{�{�NPVPR�T�H�Jbi�NON�y�fOW'T'K�W�i/a�VPV�J�R�J�H�S�a�\|NO^'T�K-j k R�J�H�S�a�\|NO^'T�Nf : Mn → Nm

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x ∈ Mnr6`�S'J�T�fO{�Jbi c�^'abd�R�J�U'W�a�n

(U, ϕ), (V, ψ)r6W�a�S�R�g {�QC^'J�LPJ�J�H�S�a�fxT�d�v

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x ∈ Unf(x) ∈ V

nbTW (x)

r-J�W'S'NPVPR�^'J�VPR�cR�J�U'W�T

xR�abW�a�d-n U�R�J

W (x) ⊂ Unf(W (x)) ⊂ V

� W�J�R�J�S�abd�VPy�o�NOVPR�{�y�NuR�{_VPT�i�y^'NO`'S�NOS'gh{�^�J�VPR�T

f{x��jn��J�L l a~JbR�J�H�S'ab\|NO^�T'N

ψfϕ−1 : ϕ(W (x)) → ψ(V )

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J�R�J�H�S�ab\|NP^'T�dRn

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f : Mn → Nm^�abfOgh{a�NuR�V�d�L¢i/a l W'T'Q

J�R�J�H�S�ab\|NP^'T'NOQ�{�J�W'S'NPVPR�^'J�VPR�TMR�J�U'W�Tx ∈ Mn

n NOV�i T@^'NPW�J�R�J�S�J�N[�öa�R�J�L l a�Ti FIH�J�N�{~VOT�i�y�L¢i/a l W�J�VuR�Tq�%y�^'W'�'T�K�VOW�i NOK'W�T � W�J�J�S l T'^�abR�^'J�N�`�S'N l VuR�a�{bi NP^'T'Nf{%J�W'S'NOVuR�^'J�VPR�T

xd'{bi�d�NPR�V�d_L¢i a l W'T'QzJ�R�J�H�S�ab\|NP^'T'NPQzJ�H�i/a�VPR�NOK¦NO{�W�i T l J�{�g(v

`'S'J�VuR�S�a�^�VPR�{�j�ºk R|i J�W�a�i c�^'g(v�J�`'S'N l Nui�NO^'T'K�`'NOS�NOK l NPQ"W�L¢i J�Ha�i�c�^'ghQhj­�®�¯ °�±3°'²�° ³�´�°Zµ�k R�J�H�S�ab\pNO^'T'N

f : Mn → NmQC^�J�LOJ�J�H�S�abfOT'K�^�abfOgh{a�NPR�r

V�d�½·¿ À�Á�ÂÃ�¶%n�NOVui�T�J�^'J~L¢i/a l W�J�N�{~J�W�S'NOVuR�^'J�VuR�TqW�ab\ l J�K�R�J�U'W'Tx ∈Mn

j19R�NOJ�S'T'TqL¢i/a l W'T�v�QC^'J�LOJ�J�H�S�abfOT'K�L¢i a l W'T'N�J�R�J�H�S�ab\|NP^'T�d�T'LOS�abFmR�Rby�\pN

S'Jbi c'n�U�R�J�W�a�W�y�F�T'LPS�a�FtR�^�NO`'S'NPS'gh{�^'ghN~J�R�J�H�S�a�\|NO^'T�dq{�J�H�o�NOKqR�NPJ�S'T'TqR�J�r`'Jbi J�LPT'U'NPVOW'T�v�`'S�J�VPR�S�a�^'VuR�{'j

�%S'J�VuR�NPK�wpT'Q[`'S�T'QCNOS'J�QqL¢i/a l W�J�LPJ%J�R�J�H�S�ab\|NP^'T�d�QCJ�\pNPR�V�i�y�\�T�R�cIR�J�\ l N�rVPR�{�NO^'^'J�N%J�R�J�H�S�ab\|NP^'T'NIL¢i a l W�J�LOJ|QC^�J�LOJ�J�H�S�abfOT�d�^'a�VONPH�d�jby%J�QC`�J�fxT'��T�d l {by�vL¢i/a l W'T�v�J�R�J�H�S�ab\|NP^'T'K�R�a�W�\|N�d'{bi'd'NPR�V�d�L¢i a l W'T'Q�J�R�J�H�S�ab\|NP^'T'NPQhj» D ê�×%í·ê�ìxß�ÙÚ×%à�Ô¢è�å'ï�ÔuãÈÔuÔuóPÖ�Ù�çÈâ�>IÕ

RNÔPà�Ö�Ý·èbÝ·ê�ÝÈï�â�Ý�ãëê�Ù�èbÛxÔPíäØ�â%ÔuØuÔuóuÖbÙÚÜ:ÝÈï�â]×�Õ Ø�ÔPß�ÛxÝå'Ô�Ühï�Ô(è�Ù�Ø�òhÕ(èbÖ�ìOãÈâ]õmØuÝäÖ�å'â�ïbÙÚõðAK�ìOí·Ø�òMn

ÞNm

î/à�Ô¢è�å'ï�ÔPãäÔPÔuóuÖbÙ�çäâ�×tÕRN

âRM

íäÔuÔuØ�îÕPÝäØuíäØ�ÕuÝÈï�ï�Ôbð���Ôuã«è�ÙmÔPØ�ÔuóPÖ�ÙÚÜ(Ýäï�â�Ýf : Mn → Nm

ï�Ù�çÈá=ÕPÙuÝäØuí«×Gãëê�Ù�èbÛ�âbå�ÔuØuÔuóuÖbÙÚÜ:ÝÈï�â�Ý¢å�ÕÔPÛ�Ö�Ý¢í·Ø�ï�ÔOí·Ø�â%ØuÔuß�Û�âx ∈ Mn

Þ�ÝÈí·ê�â�í·ìx÷3Ý¢í·Ø�ÕPìxé Ø6ÔuØ�Û�Ö�á=Ø�ÔuÝ:å'ï�Ô�Ü:Ý¢í·Ø�ÕPÔU ⊂ R

NÞx ∈ U

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Page 8: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

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Page 9: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

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k U'NP{�T l ^'J�n�LOJ�QCNPJ�QCJ�S'��T�fOQCg ^'N�VOJxv�S�a�^�d'FmR_VPR�S�y�W�R�y�S�y�L¢i/a l W�J�LOJ�QC^�J�LOJbrJ�H�S�abfOT�d��Îv�J�R]d¦W�a�S�R�a�`'Jbrº`�S'NP\_^'NOQ�y�`'NOS�NuvJ l T�Rp{�W�a�S�R�y�nL¢i/a l W�J�VuR�c���y�^'W��'T'KVOW�i NPK'W'T"S'abfxS�y�wpa�NPR�V�d �uj(�%J�sPR�J�Q�yMR�S'NPH�y�NuR�V�d"y�VOT�i�T�R�c@J�`'S'N l Nui NP^'T'N�LPJ�QCNOJ�rQCJ�S'��T�fxQ�a|R�a�W-n U�R�J�H�g L¢i/a l W�abd�VPR�S�y�W�R�y�S�a�VPJxv�S'a�^�d�i a�VOc'j

­�®�¯ °�±3°'²�° ³�´�°Zµ ��y�VuR�cf : M → N

LOJ�QCNOJ�QCJ�S��GT�fxQ L¢i a l W'T�v1QC^�J�LOJ�J�HbrS�abfxT�K-j3}�V�i T�J�R�J�H�S�ab\|NP^'T'N

fd�{bi�d�NPR�V�d�L¢i/a l W'T'Qhn�T-n�H�Jbi NON�R�J�LOJ'n�J�H�S�abR�^�J�N

J�R�J�H�S�ab\|NP^'T'Nf−1 R�a�W¡\|N�d'{�i�d'NuR�V�d�L¢i/a l W�T'Qhn�R�J[J�R�J�H�S�ab\pNO^'T'N

f^�abfxgh{�a�r

NPR�V�d Á�Ã�jj�¸OÅ�¶GÅ¢¹(jCÃ�ÇO¶�ÅÚ¶�j:}CVui T�VPy�opNPVPR�{by�NPR l T'����NOJ�QCJ�S'��T�fxQM

{Nn�R�J

LOJ�{�J�S�d�R�n U�R�J¦QC^'J�LPJ�J�H�S�a�fxT�d[Á�Ã�jj�¸OÅÚ¶�Å¢¹(jC»�¼Cj+¾\ ÀbÁ�ÀA]�À¢�(`��%S'T'{�NPVPR�T�`'S'T'QCNPS�L¢i/a l W�J�LPJpLOJ�QCNOJ�QCJ�S��GT�fxQ�a�n�W�J�R�J�S'ghK�^'N d'{br

i�d'NuR�V�d l T'����NOJ�QCJ�S��GT�fxQCJ�Qhjc3¸'d~¸P»�à ¸�`fm�a�VPVOQCJ�R�S'T'Q J�R�J�H�S�ab\|NP^'T'Np`'Sd�QCJ�K[VOJ�VuR�a�^ l a�S�R�^'J�KzVPR�S�y�W�R�yr

S'J�K�L¢i/a l W�J�LPJ�QC^'J�LPJ�J�H�S'abfxT�d�^'a~VPNOH]d-n�fxa l a�{abNOQCJ�N%��J�S'Q�y]i J�K

f : R → R f(x) = x3

z�VP^'J'nmU�R�J 8 r�L¢i a l W'T'K�LPJ�QCNOJ�QCJ�S'��T�fxQhj kml ^�a�W�J'nIJ�H�S�abR�^'J�N@J�R�J�H�S�a�\|NO^'T�Nf−1(x) = 3

√x^'N�d�{bi�d�NPR�V�d�L¢i/a l W�T'Q ��R�j W"^'N�T'QCNONuR@`'S'J�T�fx{�J l ^�J�K"{1^�y]i N]��j

��J�sPR�J�Q�yf^�N�d�{bi�d�NPR�V�d l T'����NOJ�QCJ�S'��T�fOQCJ�Qhj

��({�J�K'VPR�{a�QC^�J�LOJ�J�H�S�abfOT'K-nZVOJxv�S�a�^�d�Fmo�T'NOV�d�`'S'T l T'���GNPJ�QCJ�S'��T�fOQ�a�v1^�a�rfxgh{a�FtR�V�d1Á�Ã�jj3¸º¹/¸P»nl�Ã�ÀO¿�¿»�¼ ¶%ÃeÃ�»�#�Àu¹ Ã�Àb»n&|À]¶%Ã�j ktl ^'T'Q¡T�f%R�abW'T�v�T�^'{a�S'T�ra�^�R�J�{¦d�{bi�d'NuR�V�d�S�abfOQCNOS'^'J�VPR�c_QC^'J�LPJ�J�H�S�a�fxT�d�j�À

\ ÀbÁ�ÀA]�À���`C§-a l a l T'Q�^'at`'Sd�QCJ�K���V�J�H�ghU'^'J�K�R�J�`'Jbi J�LPT'NOK��ZVuR�S�y�W�Rby�S�ypL¢i/a l rW�J�LOJ QC^'J�LPJ�J�H�S�a�fxT�d l {by�Q�d�S'abfx^'ghQCTGVO`�J�VOJ�H�a�QCT|ËxVC`'J�QCJbopc�FMW�a�S�R

(R, ϕ(x) =x)

T(R, ψ(x) = x3)

VOJ�JbR�{�NuR�VPR�{�NO^�^'J'j)��J�W�ab\_T�R�N�n U�R�J�sPR�T QC^'J�LOJ�J�H�S�abfxT�dl T'����NOJ�QCJ�S��G^�g%j�Á ñ ÔPÕuÔPÛ�ìxà�ï�ÔPíäØ�òhãëêbÙÚèbÛ�â]õ�å'ï�ÔPãäÔuÔPóuÖ�Ùuçäâ�>%ÔPóuÖ�ÙuçäìPÝäØ�.�Úü6úREU�W��I*Ã=Þå'ÔuÖ�æ�â�çÈå�Ùuå'âGÛOÔPØ�ÔuÖ�Ô'>í·ê�ìOÜ(ÙuØ3è�â�æCæ�ÝäÔOå'ÔuÖ�æ�â�ç¢å'á�å'ï�ÔPãäÔuÔPóuÖ�Ùuçäâ�>�ðÄRÅ ï�ÙÚêbÔuãäâ�ß�ï�áL>(Ö�ÝäçÈìÈêbòÚØ�ÙuØ�ÕPÝäÖ�ÝÈï:â3è�ê�×6ãäÔOå'ÝäÔOå'ÔuÖ�æ�ï�á-õ�ØuÔuà�Ôuê�ÔuãÈâ�ß�ÝÈíäÛ�â]õ�à�Ö�ÔPí·ØuÖ�Ù�ï�íäØ�Õ�Þï�ÔhÝäãäÔ:èbÔPÛ]Ù�çÈÙuØ�Ý·êbòPí·ØuÕuÔ3ãÈÔuÖ�ÙuçºèbÔ6íºêbÔ�Ü6ï�ÝäÝuðÆOB�ØÚÙ(çÈÙÚè�Ù�ß�Ù3ÔuØ�Ö�Ù�Ü(ÙuÝäØ(Ôuóu÷:ìxé�íäâ�Ø�ìPÙ'=�â�é:ð*ÇCÙ�ÛmìPÜ(Ý�ÔuØuå'ÝÈß�ÙÚê�ÔOí·òbÞOÝ¢íºê�â�ï�Ù6å'ï�ÔPãäÔPÔuóuÖbÙÚîçÈâ�âGíäìO÷:ÝÈíäØ�ÕuìPÝäØhõOÔuØ¢×%óuáeÔ¢èbï�Ù

Cnî íäØ�Ö�ìxÛ�Ø�ìxÖ�Ù

(n ≥ 1)Þ�Ø�ÔtïbÙhï�ÝÈå�í·ìx÷3Ý¢í·Ø�ÕPìuÝäØhóPÝÈíäÛOÔPï�Ýäß�îï�Ô�å'ï�ÔuãÈÔ%ãëêbÙ�èbÛ�â]õ�íäØ�Ö�ìxÛ�Ø�ìxÖ�Þ�Ô¢èbï�Ù�ÛxÔ�Ôuà�Ö�ݺèbÝäê�×�ÝÈå'á=Ýhâbå'â|ãëêbÙ�èbÛ�â�Ý å'ï�ÔuãÈÔuÔPóuÖ�Ùuçäâ]×GóPÔ�ê�òuî

È ÝR>%ß�ÙPí·Ø�òPé�èbâ�æCæ�ÝÈÔPå'ÔuÖbæ�ï�áCð�?CÙ�à�Ö�â�å'ÝäÖ�Þ�â�çäÕPÝÈí·Øuï�Ô�Þ�ß�Ø�Ô ÕPíäÝCãëêbÙ�è�Û�â�Ý�í·ØuÖ�ìxÛ�Ø�ìxÖ�áeï�ÙRn

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c3¸'d~¸P»�à ¸�`���J�W�ab\|NPQhn�U�R�JpJ�R�J�H�S�a�\|NO^'T�Nf : R → R

n�J�`�S'N l N�i NO^�^'J�N�V�i N l yrFmopT�Q�J�H�S�abfOJ�Q

f(x) = 3√xd�{bi�d�NPR�V�d l T'����NOJ�QCJ�S��GT�fxJ�Q@y�W�abf]ab^'^'g(v�QC^�J�LOJbr

J�H�S�abfOT'K-jR

f−→ R

↓ ϕ ↓ ψR R¬�NP`'S'NPS'gh{�^'J�VuR�c

fJ�U�NO{�T l ^�a�j�y%J�J�S l T�^�abR�^�J�N�`�S'N l VuR�a�{bi NP^'T'N

fT'QCNPNPR_{�T l Ë

ψfϕ−1(x) = ψf(x) = ψ( 3√x) = x

£:R�J9L¢i/a l W�J�N�J�R�J�H�S�ab\|NP^'T'N@NP{�W�i T l J�{�g(v `'S'J�VuR�S�a�^�VPR�{�n%`'J�sPR�J�Q�y 8 r�L¢i/a l rW�J�N�J�R�J�H�S�a�\|NO^'T�N�QC^'J�LOJ�J�H�S�abfxT�K-j�y%J�J�S l T'^�a�R�^'J�N�`'S'N l VPR�a�{�i NO^'T�NpJ�H�S�a�R�^'J�LPJJ�R�J�H�S�ab\|NP^'T�d

f−1(x) = x3 T'QCNPNPR_{�T l

ϕf−1ψ−1(x) = ϕf−1( 3√x) = ϕ(x) = x

��J�sPR�J�Q�yf−1 R�J]\|N�L¢i a l W�J�NGJ�R�J�H�S�ab\|NP^'T'N�n�V�i N l J�{abR�Nui c�^�J f r l T'����NOJ�QCJ�S�r

�GT�fxQhj�1�Q�abR�NOQ�abR�T'U'NOVPW�J�Q a�^�a�i T�fxN��'NO^�R�S�a�i�c�^'J�K�T l NONPK�d�{bi�d'NuR�V�d�`'NOS�NuvJ l JbR

l T'����NOS'NP^'�'T'S�y�NOQCg(ve��y�^'W��'T'K�J�H�opNPLOJ_{�T l a|W�H�Jbi NPN�y�fOW�J�Q�y�W�i/a�VPVPy�r3QC^'JbrLOJ�U�i�NO^�a�Q���^�a]U�a�i c�^'ghQXJ�R�S�NPfxW�a�QXVPJ�J�R�{�NPR�VPR�{by�FtopNPLOJ�S�d l a��jf1 `'S'J�VuR�NOK�wpNOQVui�y�U�a�N�`'NOS�NuvJ l J�VPy�opNPVPR�{�i�d'NuR�V�d9W�VOJ�J�R�{�NPR�VPR�{�y�FmopNPK"i T'^'NPK'^'J�K���y�^'W��'T'Tr l T��G��NPS'NO^'��T�a�i�y�j/�(J�J�R�{�NPR�VPR�{�y�FmopT�K QCNPR�J l ^�abfOgh{a�NPR�V�d9¿ Ã�» ¸ÈÀu¹�Ã�Ç�ÀCl�à ¸.i�j19V�i�y�U�a�N�L¢i/a l W'T�v�QC^'J�LPJ�J�H�S'abfxT'K�n l i�d�`'S�J�T�fx{�Jbi c�^'J�LOJ~L¢i/a l W�J�LPJ_JbR�J�H�S'ab\|Nur^'T�d�R�J]\|N�QCJ]\_^'J¦`'J�VPR�S'J�T�R�c�NOVuR�NPVPR�{�NO^'^'J�N|i�T'^'NOK�^'J�N�JbR�J�H�S'ab\|NO^�T'N�j k ^'J_R�JT�^�a�fxgh{a�NuR�V�d l T'����NOS'NP^'�'T�a�i J�Qhj

��y�VuR�cF : M → N

L¢i/a l W�J�NIJ�R�J�H�S�a�\|NO^'T�N�Tx ∈ M

`'S'J�T�fx{�Jbi�c�^�abd¦R�J�U�W�a^�a

Mj/��J�L l a¡QCJ]\_^'JMJ�`'S�N l N�i T�R�c�Vui N l y�Fmo�NON�NPVPR�NOVPR�{�NO^�^'J�N@J�R�J�H�S�a�\|NO^'T�N

W�abVxabR�Nui c�^�J�LOJ_`'S'J�VuR�S�a�^�VPR�{�aTxM

{~W�a�Vxa�R�Nui�c�^'J�N�`'S'J�VuR�S�a�^�VPR�{�JTF (x)N

j

dxF : TxM → TF (x)N

ÿ Û�Ö�ÔOå'Ý n = 4Þ]ß�Ø�ÔhìÈèbâ�Õuâ�Ø�Ýäê�òuï�Ô*É���çÈÙ�è�Ù�é Ø3èbâ�æCæ�ÝÈÔPå'ÔPÖ�æ�ï�á=Ý�å'ï�ÔuãÈÔuÔuóPÖ�Ù�çÈâ]×�9xÕPíäÝ�ãëê�Ù�èbÛ�â�ÝíäØ�Ö�ìxÛ�Ø�ìxÖ�á@ï�ÙSnÞn = 1, 2, 3, 5, 6, 12

ç¢Ù�è�Ùué ØIãëêbÙ�è�Û�â�Ýhå'ï�ÔuãÈÔuÔPóuÖ�Ùuçäâ]×�ð�Ê�çäÕPÝÈí·Øuï�Ô�Ø�ÙuÛ]Ü:ÝPÞß�Ø�Ô�ÝÈíºêbâ�Ö�ÙuçÈå'ÝÈÖ�ï�ÔPíäØ�ò�å'ï�ÔuãÈÔuÔuóPÖ�Ù�çÈâ]×Gå'Ýäï�ò È Ý 4Þ�Ø�Ô�â�ç6ãÈÔPå'ÝäÔOå'ÔuÖ�æ�ï�ÔPí·ØuâGå'ï�ÔuãÈÔuÔPóuÖ�Ùuçäâ�>í·ê�ݺè�ìuÝäØ6â]õtèbâ�æCæ�ÝÈÔPå'ÔuÖbæ�ï�ÔPíäØ�ò�ð

� ÔuçÈï�â�Û]Ù�ÝÈØ�Ý¢í·Ø�Ý¢í·Ø�ÕPÝäï�ï�áL>qÕuÔuà�Ö�ÔPí'� í·ìx÷3Ý¢í·Ø�ÕPìxé Ø�ê�âqãäÔOå'ÝäÔOå'ÔuÖ�æ�ï�á=ÝPÞZï�Ô�ï�Ý|èbâ�æCæ�ÝäÔuîå'ÔPÖ�æ�ï�á=Ý=å'ï�ÔPãäÔuÔPóuÖ�Ùuçäâ]×A�LB�Ø�ÔPØ�ÕPÔuà�Ö�ÔPí óPá-ê(Ö�Ý È ÝÈï D Ü%ð@Ë%â]êbï�ÔuÖ�ÔOå�Õ�ÌMÍ�Î�ÏZãºð.��ï(à�ÔPÛxÙuçÈÙ�ê�Þß�Ø�Ô�í·ìx÷3Ý¢í·Ø�ÕPìuÝÈØ Ö�ÔPÕuï�Ô�<�Ð�ãëêbÙ�è�Û�â]õ3å'ï�ÔPãäÔuÔPóuÖ�Ùuçäâ�> ÿ íäæ�ÝÈÖ�áÑË�â]ê�ï�ÔPÖ�Ù.�·ÞÚãäÔPå'ÝÈÔPå'ÔPÖ�æ�ï�á-õ S7Þï�Ô:ï�Ý èbâ�æCæ�ÝÈÔPå'ÔPÖ�æ�ï�á-õ:èbÖ�ìxãZè�Ö�ìxãäìuð�B�Ø�Ô(ÔuØ�Û�Ö�á=Ø�â�Ý�à�Ôuê�Ô�Ühâ]ê�Ô3ïbÙ�ß�ÙÚê�Ô3Õuá�èbÝäê�Ýäï�â�é�èbâbæ�îæ�ÝÈÖ�Ýäï*=�â�ÙÚêbòuï�Ô.>mØ�ÔPà�Ô�êbÔuãäâ�âmÕhíäÙPå'ÔPíäØ�Ô�×�Ø�Ý·êbòuï�ìxé�Ôuó�ê�Ùuí·Øuò6å�ÙuØ�ÝÈå�ÙuØ�â�Û�â

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Page 11: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

Ò ³ÔÓ�u�¯ ´Õu�³ÕÖ=³Ôs�°Ñs�®�¯/°�±3°'²�° ³�´�° µ �%Jbi c�fPy�NOQCV�dGJ�`�S'N l N�i NO^�T'NOQeW�a�VOabR�N�i cbr^'J�LOJ[{�NPW�R�J�S�a�n�W�a�W¡J�`'NPS�a��'T�T l T��G��NPS'NO^'��T'S'J�{ab^'T�d-j�צ}�V�i T

v ∈ TxMnh ∈

C∞(N)r3L¢i/a l W�a�d��%y�^'W'��T�d�^'a

Nn'R�J

dxF (v)(h) = v(h ◦ F )Ø sÔs�¯�±3´�³ÕuLÖ=³Ôs�°ks�®�¯/°�±3°'²�° ³�´�° µ ��y�VPR�c(x1, . . . , xn)

T(y1, . . . , ym)

i JbrW�a�i c�^'ghN�W�J�J�S l T'^'abR�g {qJ�W'S'NOVuR�^'J�VPR]d�v�R�J�U'NOW

xTf(x)

VPJ�J�R�{�NPR�VPR�{�NO^'^'J�j:�`�y�VuR�c

yi = yi(x1, . . . , xn)ni = 1, . . . , m

rZW�J�J�S l T�^�abR�^�J�N `'S'N l VPR�a�{bi�NO^'T'NmJ�R�J�HbrS�ab\|NP^'T�d

Fjb��J�L l a

(dxF (v))i = ∂yi

∂xj vjj

Ù °�sÔt%°nÖ=¯ ´ÔÚ�°�ÛLÜ�s�°Ýs�®�¯ °�±3°'²�° ³�´�°Zµ ��y�VPR�cγ(t)

rGL¢i/a l W�abd�W'S'T�{abd�^�aM

R�a�W�abd�n�U�R�Jγ(0) = x

Tγ(0) = v ∈ TxM

j���J�L l adxF (v) = d

dt

t=0F (γ(t))1ÊU�abVPR�^�J�Q�V�i�y�U�a�N�L¢i/a l W�J�K¡�%y�^'W'�'T�T

f : Mn → R^�aqQC^'J�LPJ�J�H�S'abfxT'T�n

QCg {�W�ab\ l J�K�R�J�U'W�Nx ∈ M

T'QCNONPQ i T'^'NPK'^'ghK¡�%y�^�W'�'T'J�^'a�idxf : TxM ∼=

Rn → Tf(x)R∼= R

je�abR�S'T���a�suR�J�LPJ��%y�^'W'�'T�J�^�a�i/a�{�i J�W�a�i c�^'g(v1W�J�J�S l T'^�a�rR�a�v

(x1, . . . , xn)^'a

Mnr:sPR�J�VPR�S'J�W�a�n/VOJ�VuR�J]d�opabdeT�f�U�a�VuR�^'g(v�`'S'J�T�fx{�J l ^�g(v

∂f

∂xi

j�£:R�J�Rp�%y�^'W'�'T�J�^�a�i�^�abfOgh{a�FmR�½«¹�À�Á�à ¸u»n&~ÅÚ¶ �%y�^'W'��T'Tf{%R�J�U�W�N

xT~J�H�J�r

fx^�a]U�a�FtR�U�NOS'Nuf�Þ�v 2 3 f |x j�ß\ ÀbÁ�ÀA]�À¢��`àm�a�VOVPQCJ�R�S�T'Q9J�R�J�H�S�ab\|NP^'T'Nexp : M(n) → M(n)

n�L l NM(n) ∼=

Rn2 r3`�S'J�VPR�S�a�^'VuR�{�J�W'{a l S'abR�^'g(v�Q�a�R�S'T'��j

expX = 1 +X +X2

2!+X3

3!+ . . . á

k `�T'VxabR�c l T'���GNPS'NO^��'T�a�i@J�R�J�H�S�ab\|NP^'T�dexp

{�^�y]i Nbn-R�j�Nd0 exp : T0M(n) →

T1M(n)j

c3¸'d~¸P»�à ¸�`��3y l T'Q `�Jbi c�fOJ�{abR�c�V�d�LONOJ�QCNPR�S�T'U'NOVPW'T'Q J�`�S'N l N�i NO^�T'NOQ l T'��r�GNPS'NO^��'T�a�i/a�jâm�a�VOVPQCJ�R�S'T�Q {

M(n)W'S'T�{�y�F

γ(t)R�abW�y�F�ntU�R�J

γ(0) = 0T

ã�ä ÔuØ¢×3ôÈØ�Ô�ÔPà�Ö�Ý·èbÝ·ê�ÝÈï�â�Ý�â:ïbÙ�â�å'ÝÈï�ÝäÝ�ï�Ù�ãëê�×xèbï�Ô�Þ¢ÛxÔuï�=�Ýäà�Ø�ìOÙÚê�òPï�ÔZÔPï�Ô ×�Õuê�×�ÝäØPí«×3ï�Ù�â�óuÔ�êbÝäÝà�Ö�Ù�ÕPâ]ê�òuï�áZå-ð ñ å'ÔuØ�Ö�â�à�Ô6ôÈØ�ÔPå'ì à�ÔuÕuÔ¢è�ìhÛ�ï�â]ÜhÛ�ìâË|ð�B�ð ÇCÙ�ç¢Ù�ÖO×�ï�Ù�åQ��ú�.S*I*I¯�*�/NOI*P(P�úQ��úRG*æSAIA�R�Cç.G*����EºúR�«ù3ú·ü��CI*IàåÎðèO��ØPå'ÝäØ�â�å�Ýä÷:Ý�ÖbÙ�çuÞ�ß�ØuÔ�ãÈÖ�Ù�è�â�Ýäï�Ø�æ�ìxï�Û�=�â�â�îôäØ�Ô�ê�â�ï�ÝR>�ï�áL>(æ�ìxï�Û�=�â�Ôuï�Ù�ê�ÞÚÙ ï�Ý�ÕPÝäÛ�ØuÔuÖÛ]ÙuÛmà�ÔPß�ÝÈå'ìOî Ø�Ôhìxß�Ù�Ø(Õhå�Ù�Ø�Ý¢å�Ù�Ø�â�ß�ÝÈíäÛOÔOå_Ùuï�ÙÚêbâ�çäÝéRK�ÝÈÖ�Ýäß�â�íºê�âbå¦ï�Ý¢í·ÛxÔ�êbòuÛxÔ3ÔOí·ï�ÔPÕuï�á-õtíäÕuÔ'>bí·Ø�Õ6ôäÛ]íäà�Ôuï�ÝÈï�=�â�Ù�ê�òuï�ÔuãäÔ:ÔuØ�ÔPóuÖ�Ù�Ü:ÝÈï�â]×Ìuð�B�Ø�ÔuØ6ÖO×xè%í«õOÔ¢èbâ�Øuí«× è�ê�×�ÕPí·Ýäõmå�Ù�ØuÖ�â�=X<]ð*ê/íºê�â%å�ÙuØ�Ö�â�=�á�ÛxÔPå�å'ìxØ�â�Ö�ìOé�Ø�Þ

XY = Y XÞ]Ø�Ô

exp (X + Y ) = exp (X) exp (Y )ë ðexp (XT ) = (expX)Tì ðAí%î�ï

(expX) = e ð:ñ XÎ]ð � èbÔOí·ØÚÙ�Ø�ÔPß�ï�Ô%å�ÙÚêbÝäï�òPÛOÔ.>�ÔPÛ�Ö�Ý¢í·Ø�ï�ÔOí·Ø�âpï�ì¢ê�×pÔuØuÔuóuÖbÙÚÜ:ÝÈï�â�Ýexp

×�Õ�ê�×�ÝÈØuí«×Gè�â�æCæ�ÝäÔuîå'ÔPÖ�æ�â�ç¢å'ÔPå-ðC��óPÖ�Ù�Ø�ï�ÔuÝ�ÔuØ�ÔPóuÖ�Ù�Ü:Ýäï�â�Ý�ç¢Ù�è�ÙuÝäØuí«×�í�à�ÔPå'ÔP÷3òuéeÖO×xè�ÙlnX = (X − 1) − (X − 1)2/2 + (X − 1)3/3− . . .

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γ(0) = Xjn��J�L l a

(d0 exp)X =d

dt

t=0

exp(γ(t)) = X

��a�W'T�Q J�H�S'abfxJ�Qhn l T'����NOS'NP^'�'T�a�i@sOW�VO`�J�^'NO^��'T�a�i c�^'J�LOJ�J�R�J�H�S�ab\|NP^'T�d1{�^�y]i N`'S'N l VPR�a�{bi'd'NPR�VOJ�H�J�K�R�J]\ l NPVPR�{�NO^'^�J�N�J�R�J�H�S�a�\|NO^'T�N�j

�k VOJ�H�ghK�T�^�R�NPS'NOV�`'S'N l VPR�a�{bi'd'FmR�^'NOW�J�R�J�S'ghN�VP`'NO��T�a�i c�^�ghN�{�T l g J�R�J�H�S�a�r\|NO^'T�K-j

­�®�¯ °�±3°'²�° ³�´�°Zµ ��i/a l W�J�N�J�R�J�H�S�a�\|NO^'T�Nf : M → N

^'abfxgh{abNPR�V�dp« Å�½«¹�ha°¹ ¸u»�à ¸È¶%nCNOV�i T l T'���GNPS'NO^��'T�a�i

dxf : TxM → Tf(x)Nd'{�i�d'NuR�V�d�QCJ�^'J�QCJ�S�r

�GT�fxQCJ�Q l i�d�i FIH�J�K�R�J�U'W�Tx ∈M

� ⇔ ∀x ∈MvOx

(dxf) = 3�~�� M �uj­�®�¯ °�±3°'²�° ³�´�°Zµ �%J�LOS�y�\pNO^'T'N

f^'abfxgh{�a�NPR�V�dò#O¿/ÅZ¹ ¸u»�à ¸È¶ NOV�i T@J�^'J�d'{�r

i�d'NuR�V�d LOJ�QCNPJ�QCJ�S'��T�fOQCJ�Q ^'a¡VP{�J�K J�H�S�a�f�j Ñôó k H�S'abf[{bi J]\|NP^'T�d9^�abfOgh{a�NuR�V�d« Å�ÁP¶%» Å�½uÅ�Å]Æ�¹�À�Ç�à ¸ä¶%j

¥�V�i J�{�T'N�QCJ�^�J�QCJ�S'��^'J�VuR�T l T'����NOS'NP^'�'T�a�i/azJ�R�J�H�S�a�\|NO^'T�df : M → Na�{�R�J�Q�abR�T'U�NOVOW�T�{bi�NOU'NuR~Vui N l y�FmopNPN�VOJ�J�R�^'J�w�NO^'T'N%^'apS�abfxQCNPS'^'J�VuR�T�QC^'J�LPJ�J�Hbr

S�abfxT�K-Ë 3�~�� M ≤ 3b~�� Nj

1(VPNm`�J�LOS�y�\|NP^'T�d¦i J�W�a�i c�^�J�y�VuR�S'J�NO^'g W�a�W�VuR�a�^ l a�S�R�^'J�NI{bi J]\|NO^�T'NRn ↪→

Rmn'`'NPS'NO{�J l d�opNPN�R�J�U'W�y

(x1, . . . , xn){|R�J�U'W�y

(x1, . . . , xm, 0, . . . , 0)j

k H�S�a�f�QC^�J�LOJ�J�H�S�abfOT�d�`�S'T�`'J�LOS�y�\|NP^'T'Te^'N�J�H]d�fxa�^�H�g6R�c�QC^'J�LOJ�J�H�S�abfxT�NOQ� VOQhj�§-a l a]U�y � ��j k H�S�abf

f(M)QC^'J�LOJ�J�H�S�abfxT�d[`'S'T[{bi J]\|NP^'T'T-n-R�j N~`'J l QC^'J�LPJbr

J�H�S�abfOT'N�n'd�{bi�d�NPR�V�d�L¢i/a l W'T'Q"QC^'J�LOJ�J�H�S�abfxT�NOQhn l T'����NOJ�QCJ�S'��^'ghQMj

\ ÀbÁ�ÀA]�À³¡a`õm�abVOVOQCJbR�S'T'Q�VPNOQCNOK�VPR�{�JMJ�R�J�H�S�ab\|NP^'T'K�`�S�d�QCJ�K"{z`�i J�VPW�J�VPR�c�nf]a l ab^'^'J�N�R�a�W

fa : t 7→ (t2, t3 + at)j �%S'T�W�a�W'T�v�fx^'a�U�NO^'T�d�v�`�a�S�a�QCNuR�S�a¦J�R�J�Hbr

S�ab\|NP^'T'N�H�y l NPR_`'J�LPS�y�\|NO^�T'NOQhn/a~`'S'T�W�abW'T�v�{bi�J�\pNO^'T'NPQö|c3¸'d~¸P»�à ¸�`�e�abR�S'T���a l T��G��NPS'NO^'��T�a�i/a¦J�R�J�H�S�ab\|NP^'T�d

fa

T'QCNONuR�{�T l

dtfa =

(

2t3t2 + a

)

k R�J�H�S�a�\|NO^'T�Nfa

Hby l NPR�`�J�LOS�y�\|NP^'T'NPQ[R�J�L l aITGR�Jbi�c�W�JmR�J�L l a�n�W�J�L l aâvOxdtfa =

1l i�d�{�VON�v

t ∈ Rjà�3i N l J�{abR�N�i c�^'J�n

fa

`'J�LOS�y�\pNO^'T'N|`�S'Tq{�VON�va 6= 0

jn�pa�i NONbnW�abW~i NPLOW�J�`�S'J�{�NPS'T�R�c�n�JbR�J�H�S'ab\|NO^�T'N

fa

H�y l NuR�{�i J]\|NO^'T�NOQzR�Jbi c�W�J�`'S'Ta > 0

j}�V�i T

a < 0n�R�J�W'S�T'{abd

fa(R)T'QCNPNPR�Vxa�QCJ�`'NOS'NPVONPU'NO^'T�Nq� `�J�VPR�S'J�K�R�N_LPS�a���T'W

sPR�J�K|W'S'T'{�J�K�`'S'Ta = −1, 0, 1

l i�dpH�Jbi c�w�NOKp^�abL¢i�d l ^�J�VPR�TpR�J�LPJ'nbW�a�W�y�VPR�S'J�NP^'JsPR�J_J�R�J�H�S�ab\|NP^'T'Nx�uj

ÒF÷OÊ�ï�Ôuã«è�Ù6óPá=ÕPÙ�ÝÈØ6à�Ô�ê�ÝÈçäï�áZå�í·ê�ݺè�ìOé�÷3ÝÈÝ�Ôuà�âbíäÙ�ï�â�Ý�Õ�êbÔ�Ü:Ýäï�â�>�ð*K�ìOí·Ø�òf : M → N

ÕPçÈÙ�îâbå'ï�Ô:Ô¢è�ï�ÔuçÈï�Ù�ß�ï�ÔPÝ í ÔPóuÖ�ÙuçäÔOå�à�ÔuãäÖ�ìPÜ(Ýäï�â�ÝuÞxà�Ö�â�ß�ÝÈå|ÔPóuÖ�Ùuçf(M)

çÈÙPå'Û�ï�ìxØ3ÕNð%��Ôuã«è�Ù

fîÕuê�Ô�Ü:ÝÈï�â�ÝPð

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� #/$%�b z� ����B������'� �� 2\ ÀbÁ�ÀA]�Àø£C`�¬GabK�R�T�J�H�S�a�f_�PW�a�VxabR�Nui c�^'J�LOJ¦{�NPW�R�J�S'a��

v =

(

0 10 0

) `'J l�l NOK�rVPR�{�T'NOQ l T'���GNPS'NO^��'T�a�i/a¦sOW�VO`�J�^'NO^�R�g {|R�J�U'W�N

X =

(

λ 00 µ

) Ë

dX exp : TXM(2) → Texp XM(2)

\ ÀbÁ�ÀA]�Àù¤�`��%y�VPR�cM

r�^'NPW�J�QC`�a�W�R�^'J�N�QC^'J�LOJ�J�H�S�abfxT�N�nf : M → R

n{�f]a�r

T'QC^'J�J l ^�J�fx^�a]U'^�J�N_V~J�H�S�abfxJ�Q `'J�LPS�y�\|NO^�T'N�j�z%{bi�d�NPR�V�d[i TzJ�^'J�{bi J]\|NO^�T'NOQö|� VOS'a�{�^'T�R�N�JbR�{�NuR�VPJ_VP^'J�VOW�J�K1§]ª�

\ ÀbÁ�ÀA]�À�ú�`(z%{bi'd'NPR�V�d|i T_JbR�J�H�S'ab\|NO^�T'Nexp

`'J�LOS�y�\pNO^'T'NPQö|m�9{bi J]\|NP^'T'NOQö|

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R3 j¬�a�`'J�QC^'T'Qhn'U�R�J�{�VPN�W'S'T'{�ghNGV�QCNPR�S'T'U'NPVOW�J�K�R�J�U'W'T�fxS'NP^'T�d�J l T'^�a�W�J�{�g {Vui N l y�FmopNPQ�VOQCghV�i N�Ë�^�aqW�ab\ l J�K¡S'NOLuy]i�d'S'^�J�K�W�S'T'{�J�K ��W'S'T'{a�d¡S'NOLuy�i'd'S'^�a�nNOVui�Tq{�NOW�R�J�SqVOW�J�S�J�VPR�T1^�T'L l Np^'N�J�H�S�abopa�NPR�V�dq{�^�y�i�c��tQCJ]\_^'J�{�{�NPVPR�TeR�a�W�y�F`�a�S�a�QCNuR�S'T�f]a��'T�FÄ��^�abRby�S'a�i c�^�y�F��un=U�R�J l i T'^�a�J�R�S'NPfOW�a�W'S�T'{�J�K[QCNP\ l y�R�J�U�rW�abQCT-n�VPJ�J�R�{�NPR�VPR�{by�FtopT'QCT�fO^�a]U'NO^'T�d'QM`�a�S'a�QCNPR�S�a

t1Tt2n�{�ghU'T'V�i�d'NuR�V�d�J l T�r

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W�abWXQC^'J]\|NOVuR�{�JM^�y]i NPK"R�a�W�J�K9L¢i/a l W�J�K9�%y�^'W'�'T�TF (x1, x2, x3)

n:U�R�JM{@sPR�J�KJ�W'S'NPVPR�^�J�VPR�T ∂F

∂x3 6= 0`'J�Vui N�`'J l v�J l d�opNPKe`'NPS'NO^�y�QCNPS�a��'T'T�W�J�J�S l T'^�abR�j

Í jõ1 l J�VuR�abR�J�U'^'J¡Q�a�i J�KXJ�W'S'NOVuR�^'J�VPR�T W�ab\ l J�KXVO{�J�NOK"R�J�U'W'TΣ

f]a l a�NPR�rV�d¡W�abW�LPS�a���T'WML¢i/a l W�J�LOJ1J�R�J�H�S�ab\|NP^'T�d

x3 = f(x1, x2)`�J�Vui N�`�J l vJ l d�o�NOK

`'NOS'NP^�y�QCNOS�ab�'T'TeW�J�J�S l T�^�abR�j©�j1 l J�VuR�abR�J�U�^'JqQ�a�i J�KzJ�W'S'NPVPR�^'J�VPR�T¡W�a�\ l J�KzVP{�J�NOK1R�J�U�W'T

Σf]a l a�NPR�V�d

W�abWMJ�H�S�abf_L¢i/a l W�J�LOJqJ�R�J�H�S�ab\pNO^'T�dr : U ⊂ R2 → R3 n�L l N U rIJ�Hbi/abVPR�c�^�a

`�i J�VPW�J�VPR�TzV|W�J�J�S l T'^�a�R�a�QCT(u1, u2)

n�`'S'T'U'NPQ {�J�{�VONuveR�J�U'W�a�vqsPR�J�K1J�Hbi/a�VuR�T{�NOW�R�J�S'g

r1 = ∂r∂u1

Tr2 = ∂r

∂u2

i T'^'NPK'^'J�^�NPf]ab{�T'VOT�QCg%jÒ«Ò Ç�ÔP÷3ìxï�íäØ�ÕuÔ ç¢Ù�Û]êbé ß�ÙuÝäØuí«×GÕIØ�ÔOå-Þ�ß�Ø�Ô�â�å'Ýäï�ï�Ômà�ÔPï]×�Ø�â�Ýhå'ï�ÔPãäÔPÔuóuÖbÙ�çäâ�×�ÕPÔuçÈï�â�Û]ê�ÔmÛ]Ù�ÛÔPóuÔPóu÷3ÝÈï�â�Ý�Ö�ÝäãÈì¢ê�×�Ö�ï�Ô.>Ià�ÔuÕuÝÈÖOõxï�ÔPí·Øuâ

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ý�u·t%°�ÚÕu�³�´�°Zµ c�¸O½@h�¿b$�¹ » Å�¬.&¢¿n(W�abWXT {MV�i�y�U�a�NqW'S�T'{�g(v�nhJ�fx^'a�U'a�NPR�nhU�R�J{�J_{�VONuv�R�J�U'W�a�v�^�abw�NOKe`�J�{�NOS�v�^'J�VuR�TeVuy�opNOVuR�{�y�NPR�`'Jbi�^'J��'NP^'^'J�N�W�abVxabR�Nui c�^�J�N`'S'J�VuR�S�a�^�VPR�{�J'j��%QCNP^'^'JGsPR�J�T_J�H�NOVP`'NOU'T�{a�FmRGy�V�i J�{�T�d_R�T'`�a ∂F

∂x3 6= 0T|i T'^'NPK�r

^�abd�^'NPfxa�{�T'VPT'QCJ�VuR�c�{�NOW�R�J�S'J�{r1

Tr2j

¤IJ�R]d {�VONqR�S'T J�`'S'N l Nui�NO^'T�d sOW'{�T'{a�i NP^�R�^�g%nIQCgÄ{ l a�i�c�^'NOK�wpNOQ Hby l NOQ`'Jbi c�fOJ�{abR�c�V�d~R�S'NuR�c�T�Qhn�T�VO`'Jbi�c�fOy�Fmo�T'Qzi J�W�a�i c�^'ghN W�J�J�S l T'^'abR�g%j�£:R�J�y l J�Hbr^'J~`'S'T�S�a�VPVOQCJ�R�S'NO^'T�T�W'S'T�{�g(v�^�a|`'J�{�NOS�v�^'J�VuR�T�r�^'N%^�a l J~fxa l a�{�abR�c|W'S�T'{�y�F{

R3 T_^�a�W�i/a l gh{abR�c�^�aG^'NONma�^�a�i�T�R�T'U�NOVOW�T'NIy�V�i J�{�T�d-n�U�R�J�H�gXJ�^�a�i NP\|a�i/a�^�a`'J�{�NPSv�^'J�VPR�T�n�a l J�VuR�abR�J�U'^'J�fxa l abR�c_W'S�T'{�y�F

γ : [a, b] → U ⊂ R2

W�abW¦L¢i a l W�J�N J�R�J�H�S�ab\|NP^'T'Nh{%J�H�i/a�VPR�cUj�}�V�i T~W'S'T'{�abd|^'N(i Nu\�T�R�{%J�W'S'NPVPR�^�Jbr

VPR�T�{pJ l ^'T�QCT�T�R�NOQCT�\|Nmi�J�W�a�i c�^'ghQCT�W�J�J�S l T�^�abR�a�QCT-n'R�J~J�^�a�fxa l a�NuR�V�d�W�a�WVONOQCNPK'VPR�{�J�JbR�J�H�S'ab\|NO^�T'K¡{eR�a�W'T'N�J�Hbi/abVPR�T�j�£:R�T�J�R�J�H�S�ab\|NP^'T�d¡VPW�i NOT�{a�FmR�rV�d�{�J�H�i/a�VPR]d�v�n�W�J�R�J�S'ghN�J�`'T�VOgh{a�FtR�V�d�^'NPVOW�Jbi c�W'T'QCT�VOT�VPR�NPQ�a�QCT�W�J�J�S l T'^�a�R�nU�R�J�H�g W�J�S'S�NOW�R�^'J1fxa l abR�c[W'S'T'{�y�F T¡NPN�{�NOW�R�J�S'g VOW�J�S'J�VPR�T�jöe�g�H�y l NOQ f]a�r^'T'Q�abR�c�V�d�i J�W�a�i c�^�J�K�R�NPJ�S'T'NPK�`'J�{�NOS�v�^'J�VuR�NOK�n/`�J�sPR�J�Q�y�QCJ]\�^�JpVOU'T�R�abR�c'n U�R�J`'J�{�NPSv�^'J�VPR�c_f]a l abNPR�V�d�`'S'T�`'J�QCJbopT�J l ^'J�K�VOT'VuR�NOQCg�W�J�J�S l T'^�abRe� W�a�S�R�gt�uj

��fGJ�`'S'N l Nui�NO^'T�d�W�a�Vxa�R�Nui�c�^'J�LPJ�`'S'J�VPR�S'a�^'VPR�{a�T�R�S'NPH�J�{ab^'T�d�S'NOLuy�i'd'S'^'J�rVPR�T�Vui�N l y�NPR�nCU�R�J1{qW�ab\ l J�KzR�J�U'W�N

x = r(u1, u2)S'NOLuy�i'd'S'^'J�K�`�J�{�NOS�v�^'J�VuR�T

Σ{�NPW�R�J�S�g

r1Tr2

f]a l a�FmR¦HabfOT'V%{TxΣ

j £:R�J�R_HabfOT'V%^�abfOgh{a�NuR�V�d1ÂÀ�» Å]»�Ãn]�¸.°¬PÂÃ�¶%jLy%S'J�QCN|R�J�LPJ'n-^�a�W�a�VOabR�N�i c�^'J�Q `'S'J�VuR�S�a�^�VPR�{�N

TxΣn=S�a�VPVOQ�abR�S'T'{abNOQCJ�Q

W�abW1i T'^�NOK'^'J�N�`'S'J�VPR�S'a�^'VPR�{�J'n W�J�S'S'NPW�R�^'J�J�`'S'N l Nui NP^�a�^'NP{�ghS'J]\ l NO^'^'abd-nZ`'Jbri J]\�T�R�Nui c�^'J~J�`'S'N l Nui NP^'^�abd�W'{�a l S�abR�T'U'^�a�d���J�S�Q�a

gn�Q�abR�S'T'��a

GW�J�R�J�S'J�K�{

W�ab^'J�^'T'U�NOVOW�J�QXH�abfxT'VPN�T'QCNONuR�{�T l Ë

G =

(

〈r1, r1〉 〈r1, r2〉〈r1, r2〉 〈r2, r2〉

)

L l N�U�NOS'Nuf 〈, 〉 J�H�J�fO^�a]U'NO^'J~VuR�a�^ l a�S�R�^'J�N�NO{�W�i�T l J�{�J~`'S�J�T�fx{�N l NP^'T'N�{ R3 j­�®�¯ °�±3°'²�° ³�´�°Zµ y%{a l S�abR�T'U'^�abd"�GJ�S'Q�a

g^�abfOgh{a�NuR�V�dþ« ¸·¹�#bÅ%i Â�#bÀbÁ�¹�ÀC°

&|Ãn]�» Å%ikj�Å¢¹�¶�Å%iq`'J�{�NPSv�^'J�VPR�T�j�%NOS�{abd�W'{a l S�abR�T'U'^�abd���J�S�Q�a~`�J�{�NOS�v�^'J�VuR�Te`�J�fx{�J�i�d'NuR_{�ghU'T'Vui'd�R�c l i T�r

^'gÊT[y�L¢i�g�QCNP\ l yq{�NPW�R�J�S'a�QCT�{�W�a�VxabR�Nui c�^'J�Q `'S�J�VPR�S�a�^'VuR�{�N[��{�VOT�i�yqR�J�LOJ�nU�R�J�J�^'J�J�`�S'N l N�i�d'NuR�R�a�QXVPW�a�i�d'S'^�J�N�`'S�J�T�fx{�N l NP^'T'N]��n l i�T'^'g�W'S'T�{�g(ve^'a_`'Jbr{�NOS�v�^'J�VuR�TeT�`�i J�opa l c¦Vxa�QCJ�K�`�J�{�NOS�v�^'J�VuR�T-j

• ��y�VuR�cv, w ∈ TxΣ

{�NOW�R�J�S�a|{|W�a�VxabR�Nui c�^'J�Q�`�S'J�VPR�S�a�^'VuR�{�NGTϕv,w

r�y�LPJbiQCNP\ l y�^'T'QCT�n'R�J�L l a

|v| =√

G(v, v), cosϕv,w =G(v, w)

|v| · |w|`�S'T

v 6= 0Tw 6= 0

§ �

Page 16: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

• ��y�VuR�cγ : [a, b] → U

S'NPLPy]i�d'S�^�abd�W'S'T'{a�d¡^�aq`'J�{�NPSv�^'J�VPR�TΣ

Tγ =

dt= (u1, u2)

r�NONI{�NOW�R�J�S�VOW�J�S'J�VPR�T�{Gi J�W�a�i c�^�g(v�W�J�J�S l T�^�abR�a�v�n�R�J�L l a l i T�^�aW'S'T'{�J�K

L(γ) =

b∫

a

G(γ, γ) dt,

• }CVui TV ⊂ U

n'R�J_`�i J�o|a l c_U'a�VPR�Tr(V )

`�J�{�NOS�v�^'J�VuR�TΣS�ab{�^�a

Sr(V ) =

V

√det G du1du2

1 l T'����NOS�NO^'�'T'a�i c�^'J�K�LONPJ�QCNPR�S'T'T�`�NOS'{�y�F W'{a l S'abR�T'U�^�y�F ��J�S'Q�y~U'a�VPR�J�^�a�rfxgh{a�FtRÿ ºÂ�#bÀ�Á�¹�ÀA&~ÅÚ¶��Ú¿�¸ä¶�¸P»n&|À[Áu¿/Ã�»�¼¢ |`'J�{�NOS�v�^'J�VuR�T-jö1ÊsPR�J�Q�V�i�y�U�a�N�T'V�r`'Jbi c�fPy�FmR�R�a�W[^�abfOgh{a�NOQ�y�F �ufxa�`'T'VPc�{ l T'����NOS'NP^'�'T�a�i/a�v-�bjZ�%S'TqsPR�J�Q Vxa�Q�y�GJ�S'Q�y

gJ�H�J�fO^�a]U�a�FmRpU'NOS'Nuf

ds2 rh�PW'{�a l S�abRpsui NPQCNO^�R�a l i T�^'gG�bn'T-n�{pW�J�J�S l T�r^�abR�a�v(u1, u2)

n�fxa�`'T'VPgh{a�FtR�{¦{�T l N�Ñ �ds2 = G11(du

1)2 + G12(du1du2 + du2du1) + G22(du

2)2

}�V�i T L¢i/a l W�J�N[QC^'J�LPJ�J�H�S'abfxT'N�d'{bi�d�NPR�V�d J�H�J�H�o�NO^'T'NPQ `�J�^�d�R�T�d9S'NPLPy]i�d'S�^'J�K`'J�{�NPSv�^'J�VPR�T�n6R�J¡a�^�a�i J�LOJ�Q�`'NOS'{�J�KXW'{a l S�abR�T�U'^'J�KX��J�S'QCg `'J�{�NOS�v�^'J�VuR�T9{Vui�y�U�a�NGQC^'J�LPJ�J�H�S'abfxT'K�d'{�i�d'NuR�V�d�¹ Ã�¶�Àb» Å�#bÀ|¶�¸.&I¹�Ã�ÂÀ�j

\ ÀbÁ�ÀA]�À�^a`��pa�^[LONui�T'W�J�T l Ñôºr = (u sinϕ, u cosϕ, ϕ)

jfm�a�VPVOQCJ�R�S'T'Q W'S�T'{�Jbri T'^'NPK'^'ghK�R�S'NPy�LOJ�i c�^'T'W�^�a|LONui�T'W�J�T l N

0 ≤ u ≤ sinhϕn0 ≤ ϕ ≤ ϕ0

j ¬GabK�R�T-Ëa�j�`�i J�opa l c¦W'S'T'{�J�i T'^'NPK'^'J�LPJ_R�S�NPy�LOJbi�c�^'T'W�a6 j l i T'^'g VPR�J�S'J�^�suR�J�LPJ¦R�S'NPy�LPJbi c�^'T�W�a Ì7 j'y�L¢i�g sPR�J�LOJ~R�S'Nuy�LOJbi c�^'T'W�a�jc3¸'d~¸P»�à ¸�` ¬�a�K l NOQ�`'NOS'{by�F�W'{a l S�abR�T'U'^�y�F��GJ�S'Q�y�LON�i T'W�J�T l a�j

r1 =∂r

∂u= (sinϕ, cosϕ, 0), r2 =

∂r

∂ϕ= (u cosϕ,−u sinϕ, 1)

��J�sPR�J�Q�y�Q�abR�S'T���a¦`�NOS'{�J�K�W'{�a l S�abR�T'U'^'J�K���J�S'QCg T'QCNONuR�{�T l Ë

G =

(

1 00 u2 + 1

)

Òô��B�ØÚÙCç¢Ù�à�âbí·òCà�Ö�â�ÔuóuÖ�ÝäØÚÙ�ÝÈØ�íÈå'áZíºê�ÞOÝÈí·ê�â(èbÔPãäÔuÕPÔuÖ�â�Ø�òPí«×(ÔuóuÔPçäï�Ù�ßbÙ�Ø�ò�ß�ÝäÖ�Ýäçdu1

âdu2

ê�â]îï�ÝR>�ï�á=Ý=æ�ìxï�Û�=�â�Ôuï�Ù�ê�á~ï�Ù�Û]ÙuíÈÙ�Ø�Ýäê�òuï�ÔPå�à�Ö�ÔOí·Ø�ÖbÙ�ï�íäØ�ÕuÝTxΣ

ÞÚÔuóPÖ�Ù�çÈìxé ÷3â�Ý�Õ èbÕuÔ.>�í·ØuÕuÝäï�ï�ÔPåà�Ö�ÔPíäØ�Ö�Ù�ïbí·Ø�ÕPÝT ∗

xΣèbÕPÔ'>�íäØ�ÕuÝÈï�ï�áL>%óOÙ�çäâbí�à�Ô ÔuØuï�Ô È Ýäï�â�é1Û%ÛxÙuï�Ôuï�â�ß�ÝÈíäÛxÔPå'ìuðA��Ôuã«è�Ù6Õuá=Ö�Ù�îÜ(Ýäï�â�ÝIè�ê�×

ds2å'Ô�Ühï�Ô�ÕPÔPíäà�Ö�â�ï�âbå�Ù�Ø�òpÛ]Ù�Û�çÈÙuà�â�í·ò�Û�ÕOÙ�èbÖ�ÙuØ�â�ß�ï�Ô'>�æ�ÔuÖ�å'á¡ÕpÕPâxèbÝ%í·â�å�îå'ÝÈØ�Ö�â�ß�ï�Ô'>�ÛOÔOå'óuâ�ïbÙ�=�â�â%à�Ö�Ôuâ�çÈÕuÝ·èbÝäï�â*>�ê�â�ï�ÝM>�ï�á-õ%æ�ÔuÖ�å-ð���ï�Ù�ß�ÝÈï�â�Ý�æ�ÔPÖ�å'á

ds2ïbÙ à�ÙuÖ�ÝÕPÝäÛ�ØuÔuÖ�ÔPÕ

vâwÕuá=ß�â�íºê�×�ÝÈØuí«×mØÚÙuÛ��

ds2(v, w) = G11v1w1 + G12(v

1w2 + v2w1) + G22v2w2Òô»

http : //mathworld.wolfram.com/Helicoid.htmlhttp : //www.coolphysics.com/4d/minimal/helicoid/helicoid.htmhttp : //www.xahlee.org/surface/helicoid− catenoid/helicoid− catenoidlg1.html

§ ©

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2 j �Ii J�o|a l c~W'S'T'{�J�i T'^'NPK'^'J�LPJ_R�S�NPy�LOJbi�c�^'T'W�a�S�ab{�^�a

S =

ϕ0∫

0

sinh ϕ∫

0

√u2 + 1 dudϕ = [

fxa�QCNO^'au = sinh x] =

ϕ20 + sinh2 ϕ0

4

6 j��:R�J�S'J�^�g W�S'T'{�Jbi�T'^'NOK�^'J�LOJ�R�S'Nuy�LPJbi c�^�T'W�a~^'a�LON�i T'W�J�T l NIf]a l a�FmR�V�d�Vui N l yrFmopT�QCT�W'S�T'{�ghQCT�� ^�a�S�T'VPy�K�R�N�W'S�T'{�Jbi T�^'NOK'^�ghK�R�S'NPy�LOJ�i c�^'T'W�^�ap`�i J�VPW�J�VPR�T-nVOJ�J�R�{�NPR�VPR�{�y�FmopT�K1J�Hbi/a�VuR�T�T�fxQCNP^'NO^'T�d�i J�W�a�i c�^'g(v�W�J�J�S l T�^�abR���Ë

γ1 : t ∈ [0, ϕ0] 7→ (0, t), γ1 = (0, 1)

γ2 : t ∈ [0, sinϕ0] 7→ (t, 0), γ2 = (1, 0)

γ3 : t ∈ [0, ϕ0] 7→ (sinh t, t), γ3 = (cosh t, 1)1(ghU'T'V�i T'Q l i�T'^'g suR�T�v�W'S�T'{�g(v�j

L1 =

ϕ0∫

0

(0, 1)

(

1 00 1

) (

01

)

dt = ϕ0

L2 =

sinh ϕ0∫

0

(1, 0)

(

1 00 1 + t2

) (

10

)

dt = sinhϕ0

L3 =

ϕ0∫

0

(cosh t, 1)

(

1 00 1 + sinh2 t

) (

cosh t1

)

dt

=

ϕ0∫

0

1 + cosh2 t+ sinh2 tdt =

ϕ0∫

0

2 cosh2 tdt =√

2 sinhϕ0

7 jf1(NPS�wpT�^'g�R�S'NPy�LPJbi c�^'T�W�a�T'QCNPFmR�V�i N l y�FtopT'Npi J�W�a�i�c�^'ghNpW�J�J�S l T�^�abR�gA12 = (0, ϕ0)

nA13 = (0, 0)

nA23 = (sinhϕ0, ϕ0)

j�y�a�VOabR�N�i c�^'ghN�{�NPW�R�J�S�g WVPR�J�S'J�^�a�Q�R�S'Nuy�LPJbi c�^�T'W�a�{~sPR�T�v�R�J�U'W�a�v�T�QCNOFmR_{�T l Ë

γ1(A12) = (0,−1), γ2(A23) = (−1, 0), γ3(A13) = (1, 1)

γ1(A13) = (0, 1), γ2(A12) = (1, 0), γ3(A23) = (− coshϕ0,−1)��NO`'NPS'c~QCJ�\_^'J~{�ghU'T�Vui T�R�c�W�J�VPT'^�y�VPg y�L¢i J�{_R�S'NPy�LPJbi c�^'T�W�a

cosψ12 =

(0,−1)

(

1 00 1

) (

10

)

(0,−1)

(

1 00 1

) (

0−1

)

·√

(1, 0)

(

1 00 1

) (

10

)

= 0

§%0

Page 18: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

cosψ13 =

(1, 1)

(

1 00 1

) (

01

)

(1, 1)

(

1 00 1

) (

11

)

·√

(0, 1)

(

1 00 1

) (

01

)

=1√2

cosψ23 =

(−1, 0)

(

1 00 1 + sinh2 ϕ0

) (

− coshϕ0

−1

)

(−1, 0)

(

1 00 1 + sinh2 ϕ0

) (

−10

)

·

1√

(− coshϕ0,−1)

(

1 00 1 + sinh2 ϕ0

) (

− coshϕ0

−1

)

=1√2

k R�VPF l a�V�i N l y�NPR�n�U�R�J�y�L¢i g9`'S'T�{�NOS�w�T'^�a�vA12

nA13

nA23

S�a�{�^'g�VOJ�J�R�{�NPR�rVPR�{�NO^'^'J

π/2, π/4, π/4j�1 l T��G��NPS'NO^'��T�a�i c�^�J�K~LONOJ�QCNPR�S�T'T�`'J�{�NOS�v�^'J�VuR�NOK¦Vui�J�\_T�i/abVOc%Vui�N l y�Fmopabd

R�NOS�QCT'^'Jbi J�LOT�d�j k HbYZNOW�R�g TM{�N�i T'U'T�^'g%n�J�`'S�N l N�i�d'NPQCghN�`�NOS'{�J�KMW'{a l S'abR�T'U�r^'J�KX��J�S'QCJ�KX`�S'T'^�d�R�JM^�abfOgh{abR�c�s�i NOQCNP^�R�a�QCT¸#b»nha&I¹/¸u»�» ¸.i ½�¸OÅÚ¶�¸.&I¹�Ã�Ã�j�y{�^�y�R�S'NP^'^'NPKpLONPJ�QCNPR�S'T'T-n�^�a�`�S'T'QCNOS�nx`'S'T'^'a l i Nu\~abR l i T'^�g�W'S'T�{�g(v�nxy�L¢i�g@QCNP\~rl y�W'S'T�{�ghQCT-n�`�i Jbo|a l T�`'J�{�NOS�v�^'J�VPR�NOK-j ktl ^�a�W�J¦`'NOS�{abd�W'{a l S�abR�T'U'^�abd���J�S�rQ�a~J�`'S'N l Nui�d�NPR¦^'N%{�VPN�j�¬�a�`'S�T'QCNOS-n�{pR�NOS'QCT�^�a�v�`'NPS'{�J�K�W'{�a l S�abR�T'U'^'J�K���J�S�rQCgÊ^'Nui�c�fOdz{�ghU'T'V�i T�R�ceW'S'T'{�T�fx^�y[W�S'T'{�J�Kz^�a�`'J�{�NPSv�^'J�VPR�T�j k H�YZNPW�R�gÊTz{�Nuri T'U'T'^�g%n l i'd�J�`'S'N l Nui NP^'T�d"W�J�R�J�S'g(v�^'N l J�VuR�abR�J�U�^'JM`'NOS�{�J�K�W'{a l S'abR�T'U�^'J�K�GJ�S'QCg%n�^�a�fxgh{a�FtR|sui�NOQCNO^�R�a�QCT¥#b» ¸.d|» ¸'i�½�¸PÅÚ¶G¸'&I¹ Ã�Ã�`'J�{�NPSv�^'J�VPR�T�j:��i'd�T�vJ�`'T'VOa�^'T�d�^�a�Q�`'J�R�S'NOH�y�NPR�V�dq{�R�J�S�abd�W'{a l S�abR�T'U'^�abd���J�S'Q�a~`'J�{�NOS�v�^'J�VPR�T-j

k H�J�fO^�a]U'T'Q"U�NOS'Nufn(u1, u2)

R�abW�y�FÊ^'J�S'Q�a�i c�W�S�NOLPy]i�d�S'^'J�Kq`�J�{�NOS�v�^'J�VuR�T{�R�J�U'W�N

r(u1, u2)n�U�R�J

(r1, r2, n)r�`�Jbi J]\�T�R�Nui�c�^'J�J�S'T'NO^�R�T'S'J�{a�^'^'ghK�S�NO`'NPS�{

R3 j k ^�a~^�a�v�J l T�R�V�d�d'{�^�J¦`�J¦��J�S�Q�y�i�N�Ë

n =r1 × r2

||r1 × r2||k H�J�fx^�a]U'T�QXU'NOS'Nuf

rjk

{�NPW�R�J�S∂2r/∂uj∂uk

jà§-a l a l T'Q"{�`'S'J�T�fO{�Jbi c�^'J�K�R�J�U'W�Nx ∈ Σ

VOT'QCQCNuR�S'T'U�^�y�F�Q�abR�S'T'��yHn'`'Jbi�J�\_T'{

Hjk = 〈rjk, n〉k W�abfOgh{a�NuR�V�d-n�U�R�J�suR�TqVOJ�J�R�^'JbwpNO^�T�dqW�J�S'S'NPW�R�^�J�J�`'S'N l Nui�d�FmR�{�`�S'J�T�fO{�Jbi cbr^'J�KqR�J�U�W�N

x`'J�{�NPSv�^'J�VPR�T

Σ^�NOW�J�R�J�S�y�F W'{�a l S�abR�T'U'^�y�F���J�S'Q�y

h^�a�W�a�VOa�r

R�Nui�c�^'J�Q"`�S'J�VPR�S�a�^'VuR�{�NTxΣ

ÑF¾ jÒ Â Ç�ÔPÖ�Ö�ÝÈÛ�Ø�ï�ÔOí·Ø�òpÔuçÈï�Ù�ß�ÙuÝäØ�Þ ß�Ø�Ô|à�Ö�â�Ö�ÝäãÈì¢ê�×�Ö�ï�Ô.>�ç¢Ùuå'ÝÈï�Ý%ÛxÔuÔuÖPèbâ�ï�Ù�Ø

v1 = v1(u1, u2)Þ

§ ¼

Page 19: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

­�®�¯ °�±3°'²�° ³�´�°Zµ y�{�a l S�abR�T'U'^�a�d¡�GJ�S'Q�ah^�abfxgh{�a�NPR�V�d #*&~Å¢¹/Å%i�Â�#bÀbÁ�¹�ÀC°

&|Ãn]�» Å%ikj�Å¢¹�¶�Å%iq`'J�{�NPSv�^'J�VPR�T�j1MJ�Rxi T�U'T'N J�RG`'NOS�{�J�K~W'{a l S�abR�T�U'^'J�K¦��J�S'QCg�`'J�{�NPSv�^'J�VPR�T�n�{bR�J�S�a�d~W'{a l r

S�abR�T�U'^�abd��GJ�S'Q�a�n{�J�J�H�opN LOJ�{�J�Sd-n�^'NtJ�H]d�fxa�^�a�H�g6R�c�^'T�^'NP{�ghS'J]\ l NO^'^�J�K-n�^'T`'Jbi J]\_T�R�N�i c�^'J_J�`'S'N l Nui NP^'^'J�K�j

1(g6d�VO^'T�Q@LONPJ�QCNPR�S'T'U'NPVOW'T'K�VPQCghVui�{bR�J�S'J�K_W'{�a l S�abR�T'U'^'J�K���J�S'QCg%j���y�VuR�c

γ : [a, b] → U ⊂ R2 r→ R

3

H�T'S'NPLPy]i�d'S�^�abd�W'S'T'{a�d�Ñ À ^�a|`'J�{�NPSv�^'J�VPR�T�jr °�s�¯ °�t�u���° ³���°Zµ y%S'T�{�T�fx^'a�W�S'T'{�J�K

γ^�a%`'J�{�NPSv�^'J�VPR�T¦y l J�{bi NuR�{�J�S�d�NPR

y�S'a�{�^'NP^'T'Fk cos θ =

H(γ, γ)

G(γ, γ),

L l Nθr�y�LOJ�ieQCNu\ l y

Tnn'^'J�S'Q�a�i�d'QCT�W�W�S'T'{�J�K�T�W�`�J�{�NOS�v�^'J�VuR�T-j

��y�VuR�c_{�NOW�R�J�Svr:W�abVxabR�Nui c�^�ghK�{�NPW�R�J�S�W�`�J�{�NOS�v�^'J�VuR�Te{~R�J�U'W�N

r(u1, u2)TnrI{�NOW�R�J�SM^'J�S'Q�a�i T@W�`'J�{�NPSv�^'J�VPR�T¡{�sPR�J�K@\|N¦R�J�U'W�N�j���S�J�{�N l NOQhn U'NOS'Nuf

R�J�U'W�yr(u1, u2)

l {by�QCNPS'^�y�F�`�i J�VPW�J�VPR�c'nZ^�abR]d'^�y�Rby�F ^�a�{�NPW�R�J�S'gnTvj=�%N�r

S'NOVPNOU'NP^'T'NqsPR�J�KX`�i J�VOW�J�VPR�T9T"`'J�{�NPS�v�^�J�VPR�T�r|W'S'T�{abdγu1,u2,v

r�^�abfOgh{a�NuR�V�d» Å�¹¶%Àx¿�¿»�¼ ¶�¬O¸']�¸u»�à ¸ä¶�n J�R�{�NOU�abFmopT�Q9R�J�U'W�N

r(u1, u2)T�W�a�VxabR�Nui c�^'J�Q�y�{�NOW�r

R�J�S�yvj���f�R�NPJ�S'NPQCg e�NO^�c�N�{�g6R�NOW�a�NuR

�t²�°�±öÛàÖLÓ�´�°Zµ y%S'T'{�T�fx^�a~^'J�S�Q�a�i c�^'J�LOJ¦VONOU�NO^'T�dγu1,u2,v

S�ab{�^�a

kn(v) = ±H(v, v)

G(v, v)�+�9NPVui T�^'J�S'Q�a�i�TeWe`'J�{�NOS�v�^'J�VPR�TqTeWe^'J�S'Q�a�i c�^�J�Q�y�VONPU'NO^'T�F�VOJ�{�`'a l a�FtR�n

� − �3T'^�a]U'Nbj��a�W'T'Q J�H�S�abfOJ�Qhn�VOS'N l TX{�VPNuv�W'S'T'{�g(v�`�S'JxvJ l d�o�T�vXU�NOS'Nuf l a�^'^�y�F R�J�U�r

W�y�`'J�{�NPS�v�^�J�VPR�TeT�T'QCNOFtopT�v l a�^�^'ghK�W�a�VOabR�N�i c�^'ghK�{�NOW�R�J�Svn�^'a�T'QCNO^�c�w�y�F

W'S'T'{�T�fx^�y�H�y l NuR�T�QCNPR�c_^'J�S'Q�a�i c�^'J�N%VPNOU'NP^'T'Nbj��%QCNONuR�QCNPVPR�J_VPJ�J�R�^�J�wpNP^'T'N�Ë

k(v)| cos θ| = kn(v)

��^�J�L l a~T'QCNO^'^�J_suR�J_y�R�{�NOS\ l NO^'T�N�^�abfOgh{a�NuR¦R�NOJ�S�NOQCJ�K e�NO^'c�N�jv2 = v2(u1, u2)

å�Ù�Ø�Ö�â�=�Ù=Û�ÕOÙ�èbÖbÙ�Ø�â�ß�ï�Ô'>Cæ�ÔuÖbå'á�óPìÈèbÝäØZå'ÝÈï]×�Ø�òPí«×CØÚÙ�Û(�H(u) = JT H(v(u))J

Þã«è�ÝJî�å�Ù�ØuÖ�â�=�Ù�ÛxÔuóPâmç¢Ùuå'Ýäï�á�ÛxÔuÔPÖuè�â�ï�Ù�Ø�ðÒôÄM?CÙuà�ÔPå'â�ï�Ù�ï�â�Ý'�

•ï�Ù�ØuìOÖbÙÚê�òPï�Ô¦à�ÙuÖ�Ùuå'ÝÈØ�Ö�â�çÈÔuÕOÙ�ï�ï�Ù�× ÿ ||γ(l)|| = 1

�hÖ�ÝÈãäì¢ê�×�Ö�ïbÙÚ×�Û�Ö�â�ÕPÙÚ×γ : [a, b] → R3ïbÙ�çäá=ÕOÙ�ÝÈØuí«×¯V�I@�úREUTO�%Y'��G*����Þ]Ý¢íºê�â

d2γ/dl2 6= 0

•GA�W�OùJ�R�Cç.Ã�óuâ�Ö�ÝäãÈìÈê�×�Ö�ï�Ô'>IÛ�Ö�â�ÕuÔ'>Iï�Ùuçäá=ÕPÙuÝäØuí«×mÕuÝäÛ�Ø�ÔuÖ

nγ = d2γ

dl2/||d2γ

dl2||

•��CI�ø@IZX�G*��� ÿ à�Ö�ÔPíäØ�Ö�Ù�ïbí·Ø�ÕPÝäï�ï�Ô.>A��Û�Ö�â�ÕuÔ'>Iï�Ùuçäá=ÕPÙuÝäØuí«×mÕuÝ·êbâ�ß�â�ï�Ù k = ||d2γ

dl2||

§x¨

Page 20: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

\ ÀbÁ�ÀA]�À¨�(`qm�a�VOVPQCJ�R�S�T'Q `'Sd'QCJ�K W'S�y�LOJ�{�J�K �'T�i T'^ l S-jm�%y�VPR�cΠ

r¦`�i J�V�rW�J�VPR�c�n `�NOS'NPVONOW�a�Fto|abd�J�VOc��'T�i�T'^ l S�a�`'J l y�L¢i J�Q

αj £:R�a�`�i J�VPW�J�VPR�c�`'NOS�NOVON�r

W�abNPRe��T�i T'^ l S@`'J�^'NOW�J�R�J�S'J�Q�y1sui�i T'`'Vuy�j�¬GabK�R�TzW'S'T'{�T�fx^�y1sPR�J�LOJ�sui'i�T'`'Vxa�{R�J�U'W�N

pn�^�abT'H�Jbi NPN%y l a�i NO^�^'J�K�J�R¦NOLOJ¦�'NP^�R�S�a�j

c3¸'d~¸P»�à ¸�`=§�j��(`'J�VOJ�H'j§-a l a l T'Qq�'T�i T'^ l S|`'a�S�a�QCNuR�S'T'U�NOVOW�T_`'S�T|`'J�QCJ�o�T�i J�W�a�i c�^�g(v|W�J�J�S l T'^'abR

(ϕ, z)Ë

r = (A cosϕ,A sinϕ, z)1(ghU'T'V�i T'Q"`'NPS'{�y�F�T�{�R�J�S�y�F W'{a l S�abR�T�U'^'ghN���J�S�QCg �'T�i�T'^ l S�a�Ë

r1 =∂r

∂ϕ= (−A sinϕ,A cosϕ, 0), r2 =

∂r

∂z= (0, 0, 1)

G =

(

A2 00 1

)

n =r1 × r2

||r1 × r2||= (cosϕ, sinϕ, 0)

r11 =∂2r

∂ϕ2= (−A cosϕ,−A sinϕ, 0), r12 =

∂2r

∂ϕ∂z= 0, r22 =

∂2r

∂z2= 0

H =

(

−A 00 0

)

yGabVxabR�Nui c�^�ghK�{�NOW�R�J�S_W�sui�i T'`'Vuy_{%R�J�U'W�NpT'QCNPNPRp{�T l

γ(p) = (ξ, 0)j���a�W¦W�abW

^'J�S'Q�a�i�c�W�`�J�{�NOS�v�^'J�VuR�TM^�ab`'S�a�{bi�NO^�a�{�J�{�^'NPw�^'J�VPR�ce�'T�i�T'^ l S�a�n�a�^'J�S'Q�a�i cW�s�i'i T'`'Vuy�r3{�^�y�R�S�c'n�R�J~y�LPJbi

θ = π/2 + αj��%J�suR�J�Q�y

k(p) cos (π/2 + α) =

(ξ, 0)

(

−A 00 0

) (

ξ0

)

(ξ, 0)

(

A2 00 1

) (

ξ0

) = − 1

A

k(p) =1

A sinαÍ �(`'J�VPJ�H'j���abW�W�a�W�^�J�S'Q�a�i c�^'ghQCT�VPNOU'NP^'T�d'QCTq��T�i T'^ l S�a_d'{�i�d'FtR�V�deJ�W'S�y�\¦r^'J�VPR�T � W'S�T'{�T�fO^'g�W�J�R�J�S�g(v@QCg�fO^�a�NPQ3�¦n=R�J�y l J�H�^'J�{�J�VO`�Jbi c�fOJ�{abR�c�V�d@��J�S�rQ�y�i�J�K

k| cos θ| = kn(v)1 l a�^'^�J�Q Vui�y�U�a�N

vrIW�a�Vxa�R�Nui�c�^'ghK¡{�NOW�R�J�SMW@s�i'i T'`�VPy@{�R�J�U�W�N

pnkn(v) =

1/AnZR�j W@`�i�J�VOW�J�VuR�c'n�^�abR]d'^�y�R�a�dM^�a�{�NOW�R�J�S�a

nTv`'NPS'NOVPNOW�a�NuR[�'T�i T'^ l S@`'J

J�W'S�y�\_^'J�VPR�T-nθ = π/2+α

j�1�T�R�J�LONI`�Jbi�y�U�a�NPQ¡R�J�R�\pNmJ�R�{�NPR�Ëk(p) = 1

A sinα

j�

Í ª

Page 21: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

\ ÀbÁ�ÀA]�À¯��`�¬�a�K�R�T|`'NPS'{�y�F"T�{bR�J�S�y�F"W�{a l S�a�R�T'U'^�ghNh�GJ�S'QCgM`'J�{�NPS�v�^�J�VPR�T-nf]a l ab^'^'J�K�{_{�T l N%LOS'a���T'W�a_�%y�^'W'�'T'T

z = f(x, y)c3¸'d~¸P»�à ¸�`���a�W�abdM`'J�{�NOS�v�^'J�VuR�c�J�U'NO^'c1`'S'J�VPR�J[f]a l a�NPR�V�d�{[`�abS�a�QCNPR�S'T'U'N�rVOW�J�Q�{�T l N�Ë

r(u1, u2) = (u1, u2, f(u1, u2))j:�pa�i NON�{�VPN�`'J~J�`'S'N l Nui NP^'T'F�j

r1 =∂r

∂u1= (1, 0, f1), r2 =

∂r

∂u2= (0, 1, f2)

G =

(

1 + f 21 f12

f12 1 + f 22

)

n =(−f1,−f2, 1)√

1 + f 21 + f 2

2

r11 = (0, 0, f11), r12 = (0, 0, f12), r22 = (0, 0, f22)

H =1

1 + f 21 + f 2

2

(

f11 f12

f12 f22

)

� \ ÀbÁ�ÀA]�À���`��%J�VPR�S'J�T'Q¡`'J���y�^'W��'T'Ty = f(x)

`'J�{�NPSv�^'J�VPR�c~{�S'abopNP^'T�dz� `�Jbr{�S�abopa�NOQ@LPS�a���T'W_{�J�W'S�y�LtJ�VOT

Ox��j�¬�a�K�R�T_`'NOS�{�y�F T_{�R�J�S�y�F W'{a l S�abR�T'U'^'ghN

�GJ�S'QCg suR�J�K�`'J�{�NPSv�^'J�VPR�T�jc3¸'d~¸P»�à ¸�` �%J�{�NOS�v�^'J�VPR�c�{�S'abopNP^'T�d�QCJ]\�^'J|fxa l abR�c_`�abS�a�QCNPR�S'T'U'NPVOW'T�Ë

r(u, ϕ) = (u, f(u) cosϕ, f(u) sinϕ)

�|a�i NON�{�VPNG`�J_J�`�S'N l N�i NO^�T'F�j

r1 =∂r

∂u= (1, f ′(u) cosϕ, f ′(u) sinϕ), r2 =

∂r

∂ϕ= (0,−f(u) sinϕ, f(u) cosϕ)

G =

(

1 + (f ′)2 00 f 2

)

n =(f ′,− cosϕ,− sinϕ)

1 + (f ′)2

r11 = (0, f ′′(u) cosϕ, f ′′(u) sinϕ), r12 = (0,−f ′(u) sinϕ, f ′(u) cosϕ),

r22 = (0,−f(u) cosϕ,−f(u) sinϕ)

H =1

1 + (f ′)2

(

−f ′′ 00 f

)

Í §

Page 22: zuev/teaching/Seminar (diff geom).pdf · NuiNO^'T'N QC^'J LPJ J H Sa fxT d j  ® ¯ ° ±3°'² ° ³ ´ °Z

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v = u + 1Tv = 3 − u

^'a`'J�{�NPSv�^'J�VPR�T

x = u cos v, y = u sin v, z = u2

\ ÀbÁ�ÀA]�Àø£C`�1:ghU'T'Vui�T�R�c�{bR�J�S�y�F W'{a l S�abR�T'U'^�y�F���J�S�Q�y�R�J�S'a�j

Í�Í


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