que dans le cas général cette fonction n’est pas définie sur toute la variété

xy

p8^p(`bpvcPvR6c;`8R6x¦`bmvf4ªLNMPgj^pom&t^U`hm.w_R6vR*mvu

M

^mvc

`ypocP8R*c;`^ru¥pvxR

T x M

±§gQqpo8gcPR#fbmvgcP¤p¤^huPMPR*fbRqpomvcP.g`b^`ypocP8R*c;`^hupvx*R#po`vgvR6cu2mvgc;`y³q±§¬ª

³¦ª@LNMPRWvR6m]wPR6^hgx6^@vm8gcP`bMPfbmvdP8M#`hMPRWfhR*§R*fbR*cx*Rtu¥m8gc;`pvfhRt`hfypoc^r§mvfbQ#R6wgc!p^a`bfbpvg8M;` gcPR4^ poc¥w

`bMPRw_gj^a`ypocx*RpomvcP`hMPR4^rR8R*m_w_R4^rgjx*^NpofbRxm8c^rR6fh8R6w¹ª

LNMR¤§dPcx`hgmvc MPgjxyM xm8fhfbR6^hu2mvcwP^q`hmR4pvxyM vR4x¦`hm8f

~

xy

mo`ypocP8R*c;`!^hupvx*RvN`hMRµu2mvgc8`

y

^mv`hMPR

QqpvcPg©§m8wfhR4pvxyMPR4wpo´`hR6fpdcPg©``hgQ#RU]\`hMPRWvR6m]wPR6^hgxt^r`bpvfhgcPpo`

x

pocwqg`hM(`hMPgj^ `bpocvR*c;` 8R6x`hmvf gj^x*pvR6w(`hMR B B (

y = exp x ()

ªLNMPg^§dcx¦`bgm8c.x6pofbfhgR6^mvdP`Upm_x*pv«w_g'CR*m8Q#mvfbuPMPgj^rQ mv`ypocP8R*c;`t^rup8xR

T x M

^mvc

M

^ª

LNMPRgc;8R*fy^rRqmvN`hMPR!R»_u¥m8cPR*c;`bgpvlQqpou«`hMRm88pvfhg`hMPQ#gjx#Qqpou80

~

xy = log x (y)

x*mvfbfhR4^ru2mvc¥wP^`hm

3.3 que dans le cas général cette fonction n’est pas définie sur toute la variété.

M

®gvdPfbR( 0LNMRvR6m]wPR6^hgx6^^a`ypofh`hgcPµpo`

x

pvfhR^r`hfbg8M;`gcPR4^gc,`hMPR(R»_u¥m8cPR*cR`hgjpoQqpoupoc¥w`hMPR wPg^r`bpvcxRpomvc`bMPR*Q pofbRx*mvc^hR*fbvR4w

pvPgu2mvgc8`

(x, y)

^mo

M

2`hMR`ypocP8R*c;`W8R6x`hmvf

~

xy

^hdxyMp8^t`bMPRvR*m_w_R4^rgjxnam8gcgcP

x

^`hm

y

Mpv^pv^p

`ypocP8R*c;`vR4x¦`hm8f

~

xy

^po`

x

^ª

·¢c`hMPR#8R*cPR6fbpvx*p8^rR8C`hMPR#mv;pofbg©`bMPQ#gxQ#pvuµgj^w_R¬cR6wR*vR6fh\]MPR6fhRmvcF`bMPR#QqpocPg§mvjwFR*»_x*R*u_`

m8cµxdP`r m_xd¥^

C x

0g`g^W`hMRxm8^hdPfbRmo `hMPR^rR*`mvlu¥m8gc;`b^WMPR6fhR^rR6vR*fypovR6m]wPR6^hgx6^U^r`bpofh`hgcP(po`

x

Q#R6R`6ª.·`gj^`bMPR(x6pv^hRqmoN`hMPR!poc;`hgu¥m_w_R4pocu2mvgc8`y^m8c`hMPR(^huPMPR6fhR

S 2

MPR*fbRq`bMPR*fbRqg^poc,gc_¬cPg`a\

mvtvR6m]wPR6^hgx6^xm8cPcPR4x¦`hgcPµ`hMPR6Q ±´¬8dPfhRvo³ªÃm8c^hR6i;dPR*c;`b\8`bMPR.xd_`h Im_xd^

C x

moUpvc]\Eu2mvgc;`

x

gj^Wg©`y^pvc;`hgu¥m_w_R4pocFu2mvgc;`pvcw.`hMR`ypocP8R*c;`x*d_`r m_x*d^Ugj^W`hMPR#xgfyxRmolfypvwPgd^

π

±§¬vdPfbR8o³¦ªLNMPR

mv;pofbg©`bMPQ#gxxyMpvfr`gj^w_R¬cR6wmvc

M − C x M

^ª

WcRU^rduPu¥m;^rR4^lgc#`hMPR^rR4i;dPR*`hMpo`

M

^{w_m]R6^} cPmo``bpo8Rtxd_`h Im_xd^pocwg^ vR*m_w_R4^rgjx*pv\x*mvQ#uPR`bRv;gHªRvª

§m8ftpoc]\(x*mvdPuPR

(x, y)

^mo

M

`hMR*fbRR*»_g^r`b^Nm8cPRpocw!mvc\m8cPRvR6m]wPR6^hgxxm8cPcPR4x¦`hgcP`hMR*Q.ª

π

®gvdPfbR8(0LNMRuPjpvx*Rmo xdP`Wmo`bMPRu2mvgc;`t»¤mo

S 2

ªtÄUcFgc_¬cPg`a\.mo@vR6m]wPR6^hgx6^namvgcPgcP

x

pocw.g`b^

pvc;`hgu¥m_w_R4pocu2mvgc;`±§`am#pofbRtfbR*uPfbR6^hR*c;`hR4wqgc(fbR6w³ª·mvcPRdPc_§m8wP^Npv2vR6m]wPR6^hgx6^ m8cq`bpvcPvR6c8`^hupvx*R

T x M

_m8cPR¬cw^`hMPRx*gfyxRmofbp8w_gd¥^

π

ªLNMPRm88pofbmo`bMPQ#gxxyMpvfr`gj^w_R¬cR6wmvc

M − C x M

^ª

ER^bp¸ gcE`hMPRufhR6;gmvd¥^^rR4x¦`hgmvc¥^`hMpo``hMPR(R»_u¥m8cPR*c;`bgpv xyMpofh`pom&t^`bm¤QqpovRx*mvfbfhR4^ru2mvc¥w¤`hm

pvc]\`bpocvR*c;`qvR4x¦`bmvf#pu2mvgc;`qmo

M

pocw`hM¥p&`qfhR4xguPfbm]x6po\8@`hMR¤m88pofbg`hMPQ#gjx(xyMpvfr`qQqpo8R6^`hMPR x*mvfbfhR4^ru2mvc¥w_R*cx*RU2R`aR*R6c!`amu¥m8gc;`

(x, y)

^mo

M

poc¥wp`ypocP8R*c;`vR6x`hm8f6ªLNMPRvm;poCgj^`hm#d^hR`hMPR

^r`hfbdx`hdPfbRmovR6x`hm8ft^rup8xRmo

T x M

`hmx*pvfhfb\m8d_`tmvdPftx6pojxdPjp&`bgm8c^Ng8RR]cPm& `hmw_m#g`6ª °g`hM

`bMPfbmvdP8M`bMPRq`am¤§dPcx`hgmvc^MPgjxyM,RM¥p¸vRnad^r`^hR*R*cg`gj^u2m8^b^rgPRq§mvfd^`bmFvR6cPR*fypog-.6R#`hMPR

m8u¥R6fbpo`hgmvc^mv«pvwPw_g`hgmvcpvcw#^rd_`hfypvx`hgmvc`hmstgR6Q#pvcPcPgjpocqQqpocg©§m8wP^6ªLNMPRUw_gºxdP`a\gj^@`bmfbR*Qqpogc

m8c`hMPRQ#pvcPg§mvjw(§m8fWpvc;\!mvu2R*fyp&`bgm8c«ªLNMR§mvm&gcP#`bpvPR^rMm&t^t^rm8Q#Rpv^hgxm8u¥R6fbpo`hgmvc^tmo`hMPR

8R6x`hmvf^hupvx*R6^pvcw`bMPR*gftxm8fhfbR6^hu2mvcw_R6cxR4^gc¤sgR*QqpoccPgpvc!8R*m8QR*`hfb\0

R6x¦`bmvft^hupvx*R sgR*QqpoccPgpvc.Q#pvcPg§mvjw

~

xy = y − x xy ~ = log x (y) y = x + xy ~ y = exp x ( xy) ~ dist(x, y) = ||y − x|| dist(x, y) = || xy|| ~ x

LpvPR 0@sR*gc;`hR6fhuPfbR`yp&`bgm8c!mvpvww_g©`bgm8c¤pocw^rm8d^r`hfypvx¦`bgm8c(m8cpqsgR*QqpvcPcPgjpocQ#pvcPg§mvjw¹ª

LNMPR4^rR¥pv^hgxmvu2R*fyp&`bgm8c^g«R6cpoPRd^N`bmqvR*cR*fypog-.6RWg`hM`hMRsgR6QqpocPgjpocQqpocPg§mvjwP^Qqpoc]\

pv8mvfbg©`bMPQq^fbR6^hR*fbvR6wEdPc;`bg`hMPR6fhR§mvf`bMPR(8R6x`hmvf^ru¥pvxR4^*ª¤·¢Q#uPR*Q#R6c8`yp&`bgm8c,motmv8pvfhg`hMQgjxqpoc¥w

R*»_u¥m8cPR*c;`hgjpoxyMpvfr`y^Wgj^W`hMPRvR6\¤mol`bMPgj^vR*cR*fypog-.4p&`hgmvcF^hgcxR#povR*m8Q#R`hfbgjx*pomvu2R*fyp&`bgm8c^x*poc¥R

R*»_uPfhR4^h^hR6w!^a`ypofh`hgcP§fhm8Q `hMR6^hRmvu2R*fyp&`hgmvc¥^*ª

LNMPgj^N^rR4x¦`bgm8cmvc(`hMPRfhR6gc;`bR*fbuPfhR*`bpo`hgmvc#mo«`bMPRpv^hgxUmvu2R*fyp&`bgm8c^luPdP`b^Np`bR*fbQ p&``hMPRfhR4x*pvj^lm8c

`bMPRw_g-'2R6fhR6c;`hgjpo@QqpocPg§mvjwP^pvcwF`bMPRsgR*QqpoccPgpvc8R*m8QR*`hfb\vª·¢cE`hMPR#§m8m&gc^hR6x`hgmvc«R#g

^hR*Rt`hMPRxmvc¥^a`bfhdx`hgmvcqmvpsgR*QqpocPcgpvcQ#R`hfbgjxo8`hMPRWvR*m_w_R4^rgjx*^pocw#R*»_u¥m8cPR*c;`hgjpoCpoc¥w#m88pofbg`hMPQ#gjx

xyM¥pofh`b^§mvf`hMPRµ^hup8xR¤mo`bMPR¤`hR6c^hmvfy^#MPgxyM RFgtd¥^rR¤`bmw_R¬¥cPRµpfbg8mvfbmvd^Qqp&`bMPR*Qqp&`bgx6po

§fypoQ#R6m8fh`hmqM¥pocw_R`hMPgj^N`a\]u2RmoQqpo`hfbgx*R6^6ª

R`Ud¥^tcPm& §m_xd^Umvc.`hMR^ru¥pvxR

Sym ⁺ _n

mo@u2m8^hg©`bg8Rw_R¬¥cPg©`bR^h\;Q#Q#R`bfhgjxQqp&`bfhgjxR4^±´`bR*c^hmvfy^y³¦ªN·`b^

mvfh`hM`bmcPmo`bgx*R`hMpo`U`bMPR^hupvx*R

Sym ⁺ _n

2po`hMPm8dPvM¤p^hdP_ ¢^rR*`tmolpq8R6x¦`bmvft^hupvx*Rv¥g^UcPmo`Wp#8R6x`hmvf

^hup8xRv¹ªRvªª]`hMRcPR*;p&`hgmvc!mo@pu¥m;^rg`hgvR* w_R*¬cPg`hRQqp&`bfhg»(gj^cPmo`u2m8^hg©`bg8R ¢w_R¬¥cPg©`bRvª

LNMR8m8po;gj^`bmU¬¥cwpsgR*QqpocPcgpvcQ#R`hfbgjx g`hMgc;`bR*fbR6^r`hgcPUR6cPmvdP8MuPfbmvu2R*fh`hgR6^6ª«· ``hdfhc^m8d_``bMp&`

g`lgj^@u¥m;^h^hgPRN`bmfhR4i;dPgfbRpoc#gc;&pvfhgjpocx*RN;\`hMPR§d¥gcPR6pvf@vfbmvdPu.±HZ_R6x¦`bgm8c Wª~ª ³ªLNMg^@R6p8wP^`hmp

8R*fb\#fhR6vdPjpofQqpocPg§mvjw!^r`hfbdx`hdPfbRWMR*fbRU`bR*c^hmvfy^ g`hMc]dP«pocw!gcP¬cPg`hRR*gvR6c;&pvdPR4^pvfhR2mo`hM.p&`

pvc¤gcP¬cPg`hRwPg^r`bpvcxRmv poc]\(u2m8^hg`hgvRw_R*¬cPg`hR^r\]Q#Q#R`hfbgjxQqp&`hfbg»0`hMRxm8cPRmvu2m8^hg`hgvRw_R*¬cPg`hR

^h\]Q#QR*`hfbgxQqpo`hfbgx*R6^tgj^tfbR*up8xR6w]\.pq^rup8xRMgxyM¤Mp8^tpoc¤gc_¬cg©`bRw_R6vR*mvuQR6c;`tgc¤R4pvxyM¤mo@g`b^

n(n + 1)/2

w_gfhR4x¦`hgmvc¥^*ªV¤mvfbR*m&8R*f4&`hMR*fbRtg^mvcPRWpocwqm8cP\m8cPRU8R*m_w_R6^hgjxnamvgcPgcpoc]\`am`hR*c¥^rm8fb^6ª LNM]d^6&`bMPRt^a`bfhd¥x¦`hdfhRRNm8_`bpvgcg^lvR*fb\xm8^hR`hmpvR4x¦`hm8f@^rup8xR8&R»PxR6u_`@`hMpo``hMPRt^hupvx*RNg^@x*dPfh8R6w¹ª

½ P Â 66 ½

R`d^xm8c^rgjw_R6fU`bMPR§m8m&gcP!p8x¦`hgmvcmol`bMPRgcPR6pvfvfbmvdPu

GL n = {M ∈ M n ∗ n /det(M) 6= 0}

m8c(`bMPR`hR6c^hmvft^hupvx*R

Sym ⁺ _n

⁰

A Σ = AΣA ^T ∀A ∈ GL n

poc¥w

Σ ∈ Sym ⁺ _n

^±Io8³

LNMPgj^pvx¦`bgm8cqg^ cp&`bdPfypo\R»]`hR6cw_R4w#`hm`ypocP8R*c;` 8R6x¦`bmvfy^0g©

Γ(t) = Σ + tW + O(t ² )

g^pxdfh8R u¥pv^b^rgcPp&`

Σ

g©`bMq`bpvcPvR6c8`vR6x`hm8f

W

v`bMPR*cq`bMPRWx*dPfbvR

A Γ(t) = AΣA ^T + O(t ² )

u¥pv^b^rR4^`hfbmvdP8M

A Σ

g©`bM`bpocvR*c;`vR4x¦`bmvf

A W

^ª

®mvm&gcP * 6,Ppoc]\qgc]&pofbgjpoc;`NwPg^r`bpvcxRmvc

Sym ⁺ _n

vR6fhg¬R6^

dist(A Σ 1 , A Σ 2 ) = dist(Σ 1 , Σ 2 )

^ª

ÃMm;m;^rgcP

A = Σ ^−1/2 ₁

PRx6pocfhR4w_dxR`bMPgj^`hmpu^hR*dw_mv IcmvfbQ]m8ftw_g^r`bpvcxR`bmqgw_R6c;`hg`a\ 0

dist(Σ 1 , Σ 2 ) = dist(Id, Σ ^−/2 ₁ Σ 2 Σ ^−/2 ₁ ) = N(Σ ^−/2 ₁ Σ 2 Σ ^−/2 ₁ ).

^±I ^³

VFmvfbR*m&vR6f6&p8^`hMPRgc]&pofbgpvcxRNMp8^`hmMPm8w§mvfpoc]\`hfypoc¥^a§m8fhQqp&`bgm8c«

N

^hMPmvdPjw¥Rgc]&pofbgpvc8`@dPc¥w_R*f

`bMPRpvx`hgmvcmo`hMPRgj^rmv`hfbmvu]\qvfbmvdu

dist(Σ, Id)

gQu2m8^hR6^`bMp&`

N (Σ) = N (Σ ⁻ ¹ )

ª@LNM;d¥^*vp8m;m_wx*poc¥w_gwp&`hRNgj^`hMPRt^hdPQ¯mv¥`bMPR^hi;dpvfhR4w

mv;pofbg©`bMPQq^mo`bMPRR*gvR6c]¸pvdR6^0

N (Σ ² ) = X

i=1,n

(log(λ i )) ² .

^±Io}8³

LNMg^ 2acPm8fhQ 3vR*fbg§\(]\x*mvc^r`hfbdx`hgmvc`hMPR^r\]Q#Q#R`hfb\.pocw.u2m8^hg©`bg8R*cPR4^h^6ª

N (Σ) = 0

gQ#uPgR4^`hM¥p&`

λ i = 1

±²pvcwExmvc]8R*fy^r\P³¦«^hm(`bMPRq^rR6upofyp&`bgm8cµp&»_gmvQ¶gj^vR*fbg¬R6w¹ª5&tm&R*8R*f42R#w_m.cPm.;cm& poc]\

^hgQuRuPfbm;mv mo `hMPR`hfbgjpocP8RgcPR4i;dpog©`a\8¹MPgjxyM^hMPmvdPjwFfbR6p8w

N(Σ 1 + Σ 2 ) ≤ N (Σ 1 ) + N (Σ 2 )

R6vR6c!gRx*pvc!8R*fbg©§\g`tR»_u2R*fbgQ#R6c8`ypo\8ª

!½ _@ LNMPRF`ypocP8R*c;`!^hupvx*RFmo

Sym ⁺ _n

g^`hMPRE^rup8xRFmv^r\]Q#Q#R`hfbgjxFQqp&`bfhgjxR4^

Sym(n)

^ª

·¢c¥w_R*R4w¹2^hgcxR`bMPR8fhm8dPu¤pvx¦`bgm8c

φ A : Σ → AΣA ^T

g^UgcPR6pvf6¥g©`y^WwPR*fbg&p&`bg8RQqpou«Cw_R*cmo`hR4w

dφ A

Σ ∈ Sym ⁺ _n

g^pv^hmR4i;dPg&pvR6c8``bm

Sym(n)

^ª

- _ ²½ ½ 6

ÄUcPmo`bMPR*fNNp¸\#`hm#w_R`bR*fbQ#gcPR`bMPRgc]&pofbgpvc;`w_gj^a`ypocx*Rg^`hMfhm8dPvM(`hMPRsgR*QqpoccPgpvc!Q#R*`hfbgxvª R`d^

`ypo8RU`bMPRQm;^a`^hgQ#uPR^hx6pojpofuPfhm_w_d¥x¦`mvc(`hMPR`bpvcPvR6c;`N^hupvx*Rpo`N`hMPRgjw_R*c;`hg`a\Qqp&`bfhg»<0g

W 1

poc¥w

W 2

pofbR`bpvcPvR6c;`WvR4x¦`bmvfy^±§gIªRvª^r\]Q#Q#R`bfhgjxQqpo`hfbgx*R6^6¥cPmv`cR6xR4^h^bpofbg\.w_R¬¥cPg©`bRcmvfu¥m;^rg`hgvR¸³¦CR

wPR¬cPR`bMPR^bx*pvpvfufhm_w_dx``hm¥R`hMRN^r`bpvcwPpvfbwQqpo`hfbg©»^bx*pvpvfuPfbm_w_dx`

Σ

«R#fhR4i;dPgfbR`bMPR*gf^bx*pojpofuPfbm_w_dx¦``bm2R#gc]&pofbgjpoc;`]\.`bMPRpvx`hgmvcµmv

pvc]\`hfypoc¥^a§m8fhQqp&`bgm8c80

hW 1|W 2i = hA W 1|A W 2i A Σ

ª@LNMPgj^ ^hMPmvdw2Rt`hfbdPRtgcqupvfr`bgx*dPpvf§mvf

A = Σ ⁻ ^1/2

¥MPgjxyMµpvm&t^d^U`hmw_R¬cR`hMR^bx*pojpofUuPfhm_w_d¥x¦`Wpo`poc]\

Σ

§fhm8Q'`bMPR^bx*pvpvftuPfbm_w_dx¦`

po`N`hMPRgjw_R*c;`bg©`a\0

hW 1|W 2i Σ = hΣ ^−1/2 W 1 Σ ^−1/2 |Σ ^−1/2 W 2 Σ ^−1/2 i Id = T r(Σ ^−1/2 W 1 Σ ⁻¹ W 2 Σ ^−1/2 )

±I&~;³

$ R6m_w_R6^hgx6^m8c `hR6c^hmvf(QqpvcPg©§m8w vmvgcP,`bMPfbmvdP8M

Σ

^g©`bM `ypocP8R*c;`!8R6x¦`bmvf

W

^hMPm8dPw Mp¸vRµ`hMPR

§m8m&gcP§m8fhQ±²^hR*RkR*cPcR6x§mvfQ#m8fhRw_R`ypog^y³D0

Γ (Σ,W) (t) = Σ ^1/2 exp(tΣ ⁻ ^1/2 W Σ ⁻ ^1/2 )Σ ^1/2

Á ½ "

Â "Á

ÄW^p8R*cPR6fbpv&uPfbmvu2R*fh`a\UmoPsgR*QqpvcPcPgjpocQqpocPg§mvjwP^668R*m_w_R4^rgjx*^¹fbR6pvg/.*RlpNm_x*pv8w_g-'2R6mvQ#mvfbuPMPgj^hQ §fbmvQ

`bMPR`ypocP8R*c;`W^hupvx*Rp&`pqvgvR6c¤u¥m8gc;`mo@`hMRQqpocPg§mvjw`hm(`hMPRQqpocPg§mvjw<0

Γ (Σ,W) (1) = exp Σ (W )

p8^h^hm_xgjp&`bR6^`bm(R4pvxyM¤`ypocP8R*c;`UvR4x¦`bmvf

W ∈ T Σ Sym ⁺ _n

p(u¥m8gc;`mo@`bMPRQqpvcPg©§m8w«ªLNMPgj^QqpouPuPgcP!g^

x6poR4w#`hMRWR*»]u2mvcR*c;`hgjpoCQqpou«]2R6x6pod^hRUg`Nx*mvfbfhR4^ru2mvc¥wP^@`bm`bMPRWd¥^rdpv¥R*»_u¥m8cPR*c;`hgjpo2gc!^hmvQ#RWQqpo`hfbg©»

8fhm8dPu^6ªLNMg^gj^NR»Pp8x¦`h\m8dPftx*p8^rR§mvfN`bMPRR»_u2mvcPR6c;`hgjpo¹Qqpou.pofbmvdPc¥wq`hMRgw_R6c;`hg`a\ 0

&Um&R6vR*f4¸`hMRtsgR6QqpocPcPgjpocR»_u2mvcPR6c;`hgjpo_Qqpouqpv^b^hm]x*gpo`hR4w`hmm8dPfgc]&pofbgjpoc;`@QR*`hfbgxMp8^lpQ#mvfbR

x*mvQ#uPR»(R*»_uPfhR4^h^hgmvc!po`mo`hMR*fN`hR6c^hmvfy^0

exp Id (U DU ^T ) = exp(U DU ^T ) = U exp(D)U ^T .

^±Ivv³

&Um&R6vR6f6C`bMPR(sgR*QqpoccPgpvcR*»_u¥m8cPR*c;`hgjpo Q#pvuEp8^h^hm_xgjp&`bR6wµ`hm¤mvdfgc]&pofbgpvc;`Q#R*`hfbgx#Mp8^p.Q#mvfbR

x*mvQ#uPR»(R*»_uPfhR4^h^hgmvc!po`mo`hMR*fN`hR6c^hmvfy^0

exp Σ (W ) = Σ ^1/2 exp(Σ ⁻ ^1/2 W Σ ⁻ ^1/2 )Σ ^1/2 .

^±Io|8³

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Dans le document An automatic mesh coarsening technique for three dimensional anisotropic meshes (Page 48-56)

que dans le cas général cette fonction n’est pas définie sur toute la variété

xy

M

T x M

~

xy

y

x

y = exp x ()

T x M

M

~

xy = log x (y)

3.3 que dans le cas général cette fonction n’est pas définie sur toute la variété.

M

x

(x, y)

M

~

xy

x

y

~

xy

x

C x

x

S 2

C x

x

π

M − C x M

M

(x, y)

M

π

S 2

x

T x M

π

M − C x M

M

(x, y)

M

T x M

~

xy = y − x xy ~ = log x (y) y = x + xy ~ y = exp x ( xy) ~ dist(x, y) = ||y − x|| dist(x, y) = || xy|| ~ x

Sym + n

Sym + n

n(n + 1)/2

GL n = {M ∈ M n ∗ n /det(M) 6= 0}

Sym + n

A Σ = AΣA T ∀A ∈ GL n

Σ ∈ Sym + n

Γ(t) = Σ + tW + O(t 2 )

Σ

W

A Γ(t) = AΣA T + O(t 2 )

A Σ

A W

Sym + n

dist(A Σ 1 , A Σ 2 ) = dist(Σ 1 , Σ 2 )

A = Σ −1/2 1

dist(Σ 1 , Σ 2 ) = dist(Id, Σ −/2 1 Σ 2 Σ −/2 1 ) = N(Σ −/2 1 Σ 2 Σ −/2 1 ).

N

dist(Σ, Id)

N (Σ) = N (Σ − 1 )

N (Σ 2 ) = X

i=1,n

(log(λ i )) 2 .

N (Σ) = 0

λ i = 1

N(Σ 1 + Σ 2 ) ≤ N (Σ 1 ) + N (Σ 2 )

Sym + n

Sym(n)

φ A : Σ → AΣA T

dφ A

Σ ∈ Sym + n

Sym(n)

W 1

Sym ⁺ _n

Sym ⁺ _n

Sym ⁺ _n

A Σ = AΣA ^T ∀A ∈ GL n

Σ ∈ Sym ⁺ _n

Γ(t) = Σ + tW + O(t ² )

A Γ(t) = AΣA ^T + O(t ² )

Sym ⁺ _n

A = Σ ^−1/2 ₁

dist(Σ 1 , Σ 2 ) = dist(Id, Σ ^−/2 ₁ Σ 2 Σ ^−/2 ₁ ) = N(Σ ^−/2 ₁ Σ 2 Σ ^−/2 ₁ ).

N (Σ) = N (Σ ⁻ ¹ )

N (Σ ² ) = X

(log(λ i )) ² .

Sym ⁺ _n

φ A : Σ → AΣA ^T

Σ ∈ Sym ⁺ _n

A = Σ ⁻ ^1/2

hW 1|W 2i Σ = hΣ ^−1/2 W 1 Σ ^−1/2 |Σ ^−1/2 W 2 Σ ^−1/2 i Id = T r(Σ ^−1/2 W 1 Σ ⁻¹ W 2 Σ ^−1/2 )

Γ (Σ,W) (t) = Σ ^1/2 exp(tΣ ⁻ ^1/2 W Σ ⁻ ^1/2 )Σ ^1/2

W ∈ T Σ Sym ⁺ _n

exp Id (U DU ^T ) = exp(U DU ^T ) = U exp(D)U ^T .

exp Σ (W ) = Σ ^1/2 exp(Σ ⁻ ^1/2 W Σ ⁻ ^1/2 )Σ ^1/2 .