UNIT 7. THE FUNDAMENTAL EQUATIONS OF HYPERSURFACE THEORY

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1 UNIT 7. THE FUNDAMENTAL EQUATIONS OF HYPERSURFACE THEORY ================================================================================================================================================================================================================================================ Gau frame of a arameterzed hyerurface, formuae for the arta dervatve of the Gau frame vector fed, Chrtoffe ymbo, Gau and Codazz-Manard equaton, fundamenta theorem of hyerurface, "Theorema Egregum", comonent of the curvature tenor, tenor n near agebra, tenor fed over a hyerurface, curvature tenor Now we derve ome formuae for hyerurface. Conder a reguar arameterzed hyerurface r:w l R. The arta dervatve r,...,r 1 n defne a ba of the tangent ace of the hyerurface at each ont. If we add to thee vector the norma vector of the hyerurface, we get a ba of R at each ont of the hyerurface. The ytem of the vector fed r,...,r,n aong r caed the Gau frame of the hyerurface. Gau 1 n frame ay mar roe n the theory of hyerurface a Frenet frame doe n curve theory. Smarty not comete however, nce a Gau frame much more deendent on the arameterzaton. Neverthee, n the ame way a for Frenet frame, t mortant to now how the dervatve of the frame vector fed wth reect to the arameter can be exreed a a near combnaton of the frame vector. For th we have to determne the coeffcent G, a, b, g n the exreon j j r = S G r + a N, N = S b r + g N. () j j j Let u begn wth the me obervaton that nce N nown to be j tangenta, and N = -L(r ), where L the Wengarten ma, j j u o g = 0 for a j j m n -1 and (-b ) the matrx L = B G of the Wengarten ma wth reect to j j,=1 the ba r,...,r. Denote by g and b the entre of the frt and 1 n econd fundamenta form a uua, and denote by g the comonent of the nvere matrx of the matrx of the frt fundamenta form. (Attenton! -1 Entre of G and G are dtnguhed by the oton of ndce.) Then u o b = - S b g. j j m Tang the dot roduct of the frt equaton of () wth N we gan the 1

2 equaty <r,n > = a and nce <r,n > = b, u o m a = b for a,j. There ony one queton eft: what are the coeffcent G equa to? Let u tae the dot roduct of the frt equaton of () wth r <r,r > = S G <r,r > = S G g, or denotng the dot roduct <r,r > horty by G, G = S G g. The coeffcent G and G are caed the Chrtoffe ymbo of frt and econd tye reectvey. The at equaton how how to exre Chrtoffe ymbo of econd tye wth the he of Chrtoffe ymbo of frt tye. It can ao be ued to exre Chrtoffe ymbo of frt tye n term of econdary Chrtoffe ymbo. Indeed, mutyng the equaton wth g, tang um for and ung S g g = d (J Kronecer deta), we get S G g = S S G g g = S G d = G. Now et u try to determne Chrtoffe ymbo of econd tye. Dfferentatng the equaty g = <r,r > wth reect to the -th varabe j and then ermutng the roe of ndce,j, we get the equate g, = <r,r > + <r,r > j j g j, = <r,r > + <r,r > j j g,j = <r,r > + <r,r >. j Sovng th near ytem of equaton for the econdary Chrtoffe ymbo tandng on the rght hand de, we obtan and 1 G = <r,r > = (g + g - g ) 2,j j,, u o 1 G = S G g = S (g + g - g ) g. 2,j j,, m Oberve that the Chrtoffe ymbo deend ony on the frt fundamenta form of the hyerurface. 2 Now we a the foowng queton. Suoe we are gven 2n mooth functon g, b,j=1,2,...,n on an oen doman W of R. When can we fnd a arameterzed hyerurface r:w l R wth fundamenta form G = (g ) and B = (b ). We have ome obvou retrcton on the functon g and b. Frt, g = g, b = b, and nce G the matrx of a otve j j defnte bnear form, the determnant of the corner ubmatrce (g ),j=1 2

3 mut be otve for = 1,...,n. However, the exame we have how that thee condton are not enough to guarantee the extence of a hyerurface. For exame, f G the dentty matrx everywhere, whe B = f G for ome functon on W, then the hyerurface (f ext) cont of umbc. We now however that f a urface cont of umbc, then the rnca curvature are contant, o athough our choce of B and G atfe a the condton we have ted o far, t doe not correond to a hyerurface une f contant. So there mut be ome further reaton between the comonent of B and G. Our an to fnd ome of thee correaton the foowng. Let u exre r and r a a near combnaton of the Gau frame vector. The coeffcent we get are functon of the entre of the frt and econd fundamenta form. For r = r, the correondng coeffcent n the exreon for thee vector mut be equa and t can be hoed that th way we arrve at further non-trva reaton between G and B. Th wa the hoohy, and now et u get down to wor. r = ( S G r + b N ) = S (G r + G r ) + b N + b N =,,, = S (G r + G (S G r + b N)) + b N - b SS b g r =,, = S (G + S G G - b S b g ) r + (b + S G b )N.,, Comarng the coeffcent of r n r and r, we obtan u o G - G + S (G G - G G ) = S (b b - b b ) g,,j j j m whe comaron of the coeffcent of N gve u o b - b = S G b - S G b.,,j j m The frt n equaton (we have an equaton for a,j,,), are the 3 Gau equaton for the hyerurface. The econd famy of n equaton are the Codazz-Manard equaton Exerce. Exre the econd order dervatve N and N a a near j combnaton of the Gau frame vector. Comare the correondng coeffcent and rove that ther equaty foow from the Gau and Codazz-Manard equaton. The exerce ont out that a mar try to derve new reaton between G and B doe not ead to reay new reut. Th no wonder, nce the Gau and Codazz-Manard equaton together wth the revouy ted obvou condton on G and B form a comete ytem of neceary and 3

4 uffcent condton for the extence of a hyerurface wth fundamenta form G and B. n Theorem. (Fundamenta theorem of hyerurface). Let W C R be an oen n connected and my connected ubet of R (e.g. an oen ba or cube), and uoe that we are gven two mooth n by n matrx vaued functon G and B on W uch that G = (g ) and B = (b ) agn to every ont a ymmetrc matrx, G gve the matrx of a otve defnte bnear form. In th cae, f the functon G derved from the comonent of G accordng to the above formuae atfy the Gau and Codazz-Manard equaton, then there ext a reguar arameterzed hyerurface r:w l R for whch the matrx rereentaton of the frt and econd fundamenta form are G and B reectvey. Furthermore, th hyerurface unque u to rgd moton of the whoe ace. Namey, f r and r are two uch hyerurface, then there 1 2 ext an ometry (=dtance reervng becton) F:R L R for whch r = Fqr. 2 1 Let u denote the exreon tandng on the eft hand de of the Gau equaton by u o R := G - G + S (G G - G G ).,,j j m Then Gau equaton can be abbrevated wrtng R = S (b b - b b ) g. j Let u muty th equaton by g and tae a um for m S R g = S S (b b - b b ) g g = m j m = S (b b - b b ) d = (b b - b b ). j m m jm u o 1 1 mj m m u o Introducng the functon R := S R g, we may wrte R = (b b - b b ). mj m jm m Let u oberve, that the functon R can be exreed n term of the mj frt fundamenta form G. Coroary. (Theorema Egregum) The Gauan curvature of a reguar arameterzed urface n R can be exreed n term of the frt fundamenta form a foow R 1212 K = det G Theorema Egregum one of thoe theorem of Gau he wa very roud of. 4

5 The urrng fact not the actua form of th formua but the mere extence of a formua that exree the Gauan curvature n term of the frt fundamenta form. The geometrca meanng of the extence of uch a formua that the Gauan curvature doe not change when we bend the urface (athough rnca curvature do change n genera!). Defnton. Let r:w l R be a hyerurface. Conder the mang R that agn four tangenta vector fed X = S X r, Y = S Y r, Z = S Z r, W = S W r a functon accordng to the formua m j R(X,Y;Z,W) = S S S S R X Y Z W. mj m j We ha ca R the curvature tenor of the hyerurface, the functon R mj the comonent of the curvature tenor Let u brefy reca ome defnton from near agebra, concernng tenor. Let V be a vector ace (over R). The et V of near functon form a vector ace wth reect to the oeraton ( + )(v) := (v) + (v), ( )(v) := ((v)) The vector ace V of near functon on V caed the dua ace of V If V fnte dmenona and e,...,e a ba of V, then we may 1 n 1 n conder the near functon e,...,e e V defned by e (e ) = d. It j j not dffcut to rove that thee near functon form a ba of V caed the dua ba of the ba e,...,e. A a conequence we get that dm V = n dm V for fnte dmenona vector ace. A tenor of vaency (/order /tye) (,) over V a mutnear functon T : V x...xv x V x...x V L R defned on the Cartean roduct of coe of V and coe of V. "Mutnear" mean that fxng a but one varabe, we obtan a near (,) functon of the free varabe. Denote by T V the et of tenor of vaency (,). The um of two tenor of order (,) and the caar mute of a tenor are tenor of the ame order, hence the et of tenor of a gven vaency form a vector ace. If e,...,e a ba of V, then every 1 n tenor T unquey determned by t vaue on ba vector combnaton,.e. by the number T = T(e,...,e ;e,...,e ), j...j j j 1 1 whch are caed the comonent of the tenor T wth reect to the ba

6 ... (+) 1 e,...,e. Snce any (dm V) number T correond to a tenor, 1 n j...j 1 (,) (+) dm T V = (dm V). Now et u conder a reguar arameterzed hyerurface M, r:w L R A tenor fed of vaency (,) over M a mang T that agn to every ont u e W a tenor of vaency (,) over the tangent ace of M at r(u). T(u) unquey determned by t comonent T (u) wth reect j...j to the ba r (u),...,r (u). The functon u LT (u) are caed the n j...j comonent of the tenor fed T. T ad to be a mooth tenor fed f t comonent are mooth. Exame. - Functon on M are tenor fed of vaency (0,0). - Tangenta vector fed are tenor fed of vaency (1,0) (V omorhc to V n a natura way). - The frt and econd fundamenta form of a hyerurface are tenor fed of vaency (0,2). - The mang that agn to every ont of a hyerurface the Wengarten ma at that ont a tenor of vaency (1,1). (The near ace of (1,1) V L V near mang omorhc to T V n a natura way.) - Let f be a mooth functon on M. Conder the tenor fed of vaency (0,1) defned on a tangent vector X to be the dervatve of f n the drecton X. Th tenor fed the dfferenta of f. - The curvature tenor a tenor fed of vaency (0,4). The curvature tenor one of the mot bac object of tudy n dfferenta geometry. In the revou comutaton the curvature tenor came acro e a rabbt from a cynder. To undertand t rea meanng, we ha ntroduce the curvature tenor n a more natura way n a more genera framewor, n the framewor of Remannan manfod. For th uroe, we have to get acquanted wth ome fundamenta defnton and contructon. Th w be the goa of the foowng unt. 6