Unit 5. Hypersurfaces

Transcription

1 Uit 5. Hyersurfaces ================================================================= Vector fields alog hyersurfaces, tagetial vector fields, derivatios of vector fields with resect to a taget directio, the Weigarte ma, biliear forms, the first ad secod fudametal forms of a hyersurface, ricial directios ad ricial curvatures, mea curvature ad the Gaussia curvature, Euler s formula Defiitio. Let r:w l R be a arameterized hyersurface. A vector field alog the hyersurface is a maig X:W L T R from the domai of arameters * ito the taget budle of R such that X(u) e T R for ay u e W. X is a r(u) tagetial vector field, if X(u) is taget to the hyersurface at r(u) ~ ~ Sice X(u) has the form (r(u),x(u)), where X:W L R, there is a oe to oe corresodece betwee vector fields alog a arameterized hyersurface ad smooth maigs of the domai of arameters ito R. Roughly seakig, if we are give a smooth maig of the arameter domai ito R, we may thik of it as a vector field alog the hyersurface though formally it is ot a vector field. I this way, the maigs r,...,r ad N should be thought 1-1 of as vector fields alog the hyersurface, the first -1 of which are tagetial. Give a vector field alog a hyersurface, we would like to exress the seed of chage of the vector field vectors as we move alog the surface, i terms of the seed of our motio. This is achieved by the followig. Defiitio. Let r:w l R be a arameterized hyersurface, X:W L R be a vector field alog it, u e W, v a taget vector of the hyersurface at 0 r(u ). We defie the derivative d X of the vector field X i the directio v v as d X = (Xqu) (0), where u: [-1,1] L W is a curve i the arameter domai v such that u(0) = u ad (rqu) (0) = v. 0 Sice by the chai rule -1 (Xqu) (0) = S u (0) X (u(0)), i i where (u,...,u ) are the comoets of u, X,...,X are the artial derivatives of X, ad by -1 v = (rqu) (0) = S u (0) r (u(0)) i i 1

2 the umbers u (0),...,u (0) 1-1 are the comoets of v i the basis r (u ),...,r (u ) of the taget sace at r(u ), 0 we have the followig formula -1 d X = S v X (u ), v i i 0 where v,...,v 1-1 are the comoets of the vector v i the basis r (u ),..., 1 0 r (u ) This formula shows that the defiitio of d X v is correct, i.e. ideedet of the choice of the curve u(t). We shall cosider the local behavior of curvature o a hyersurface. The way i which a hyersurface curves aroud i R is closely related to the way the ormal directio chages as we move from oit to oit. Lemma. The derivative d N of the ormal directio o a hyersurface with v resect to a taget vector v at = r(u) is taget to the hyersurface at r(u). Proof. We eed to show that d N is orthogoal to N(). Ideed, Y differetiatig the relatio 1 _ < N, N >, we get 0 = < d N, N > + < N, d N > = 2 < d N, N > v v v Defiitio. Let us deote by M the arameterized hyersurface r:w l R ad by T M the liear sace of its taget vectors at = r(u ). 0 The liear ma defied for a fixed e M by L : T M L T M L (v) = - d N v is called the Weigarte ma or shae oerator of M at Before goig o with the study of hyersurfaces, let us recall some defiitios from liear algebra. Defiitio. Let V be a vector sace. A biliear fuctio or form o V is a maig satisfyig the idetities B : VxV L R B(ax +bx,y) = a B(x,y) + b B(x,y), B(x,ay +by ) = a B(x,y ) + b B(x,y ) for ay x,x,x,y,y,y ev, a,ber B is said to be symmetric if B(x,y) = B(y,x) for all x y e V. A symmetric biliear fuctio is ositive defiite if B(x,x) >0 for x$ A vector sace equied with a ositive defiite symmetric biliear form is a Euclidea vector sace

3 For examle, R with the usual dot roduct o it is a Euclidea vector sace. If x,...,x is a basis of the vector sace V ad B is a biliear fuctio 1 o V the the x matrix (b ) with etries b = B(x,x ) is called the ij ij i j matrix reresetatio of B with resect to the basis x,...,x. Fixig the basis we get a oe to oe corresodece betwee biliear fuctios ad x matrices. A biliear form is symmetric if ad oly if its matrix reresetatio with resect to a basis is symmetric. Defiitio. The quadratic form of a biliear fuctio B is the fuctio defied by the equality Q (x) = B(x,x). B Symmetric biliear fuctios ca be recovered from their quadratic forms with the hel of the idetity 1 B(x,y) = (Q (x+y) - Q (x) - Q (y)). 2 B B B Now we retur to hyersurfaces. We defie two biliear forms o each taget sace of the hyersurface Defiitio. Let M be a arameterized hyersurface r:w l R, u e W, T M the liear sace of taget vectors of M at = r(u), L :T M L T M the Weigarte ma. The first fudametal form of the hyersurface is the biliear fuctio I o T M obtaied by restrictio of the dot roduct oto T M I (v,w) = < v, w > for v,w e T M. The secod fudametal form of the hyersurface is the biliear fuctio II o T M defied by the equality II (v,w) = < L v, w > for v,w e T M. The first fudametal form is obviously a ositive defiite symmetric biliear fuctio o the taget sace. Its matrix reresetatio with resect to the basis r (u ),...,r (u ) has etries < r (u ), r (u ) > i 0 j 0 A imortat roerty of the Weigarte ma ad the secod fudametal form is stated i the followig theorem. Theorem. The secod fudametal form is symmetric, i.e < L v, w > = < v, L w > for v,w e T M, or i other words, the Weigarte ma is self-adjoit (with resect to the first fudametal form). Proof. It is eough to rove that the matrix of the secod fudametal form with resect to the basis r (u ),...,r (u ) is symmetric Lemma. II (r (u ),r (u )) = < r (u ), N(u ) > i 0 j 0 ij 0 0 Proof of Lemma. We kow that the ormal vector field N is eredicular to

4 ay tagetial vector field, thus < N, r > _ 0. j Differetiatig this idetity with resect to the i-th arameter we get from which < d N, r > + < N, r > _ 0, r j ji i < N, r > _ < - d N, r > = < L r, r > ji r j r i j i Sice by Youg s theorem r = r, the lemma shows that the matrix of the ij ji secod fudametal form is symmetric Comarig the idetity roved i the lemma with the formula exressig the ormal curvature of the hyersurface i a taget directio v we see that the ormal curvature is the quotiet of the quadratic forms of the secod ad first fudametal forms -1-1 S S <N(u ),r (u )>v v 0 ij 0 i j II (v,v) j=1 k(v) = = , 2 I (v,v) v -1 where v = S v r (u ) is a taget vector of the hyersurface at = r(u ). i i 0 0 The exressio u o II (v,v) k(v) = I (v,v) m gives rise to a liear algebraic ivestigatio of the ormal curvature. It is atural to ask at which directios the ormal curvature attais its extrema. Sice k(lv) = k(v) for ay l $ 0, it is eough to cosider this questio for the restrictio of k oto the uit shere S i the taget sace. The uit shere of a Euclidea sace is a comact (=closed ad bouded) subset, thus by Weierstrass theorem, ay cotiuous fuctio defied o it attais its maximum ad miimum. Defiitio. Let f be a differetiable fuctio defied o the uit shere S of a Euclidea vector sace. We say that the vector v e S is a critical oit of f if for ay curve g : [-1,1] L S o the shere such that g(0) = v the derivative of the comosite fuctio fqg vaishes at 0. Clearly, local miimum ad maximum oits of a fuctio are its critical oits, but the coverse is ot true. The followig roositio gives a characterizatio of critical oits for the restrictio of the ormal curvature oto the uit shere of the taget sace. 4

5 Proositio. Let V be a fiite dimesioal vector sace with a ositive defiite symmetric biliear fuctio <,> ad let L:V L V be a self-adjoit liear trasformatio o V. Let S = {x e V : <x,x> = 1} ad defie f:s l R by f(x)=<lx,x>. The v e S is a critical oit of f if ad oly if v is a eigevector of L. Proof. For ay curve g:[-1,1] l S such that g(0) = v, we have d <L(g(t)),g(t)>1 = <L(g (0)),g(0)>+<L(g(0)),g (0)> d t t=0 = <Lg (0),v>+<Lv,g (0)> = 2 <Lv,g (0)>. This meas that v is a critical oit of f if ad oly if Lv is orthogoal to every vectors of the form g (0). Sice the seed vectors g (0) of sherical curves through v = g(0) rage over the taget sace of the shere S, v is a critical oit of f if ad oly if Lv is orthogoal to the taget sace of S at v however, sice the ormal vector of this taget sace is v, the latter coditio is satisfied if ad oly if Lv is a scalar multile of v, i.e. v is a eigevector of L As a alicatio of the roositio, let us rove the followig theorem of liear algebra. Theorem. Let V be a fiite dimesioal Euclidea vector sace ad let L : V L V be a self-adjoit liear trasformatio o V. The there exists a orthoormal basis of V cosistig of eigevectors of L. Proof. By iductio o the dimesio of V. For = 1 the theorem is trivial. Assume that it is true for = k. Suose = k + 1. By the roositio, there exists a uit vector v i V which is a eigevector of L. 1 1 Let W = v = {w e V : v 1 w }. The L(W) c W sice we have 1 1 <Lw,v > = <w,lv > = <w,l v > = l <w,v > = for ay w e W, where l is the eigevalue belogig to v. Clearly L1 is 1 1 W self-adjoit. Sice dim(w) = dim(v) - 1 = k, the iductio assumtio imlies that there exists a orthoormal basis (v,...,v ) for W cosistig of 2 eigevectors of L1. But each eigevector of L1 is a eigevector of L, so W (v,...,v ) is a orthoormal basis for V cosistig of eigevectors of L Defiitio. For a hyersurface M i R arameterized by r, r(u ) = e M, the eigevalues k (),...,k () of the Weigarte ma L :T M LT M are 1-1 called the ricial curvatures of M at, the uit eigevectors of L are called ricial curvature directios If the ricial curvatures are ordered so that k ()<k ()<...<k (), the discussio above shows that k () is the maximal, k () is the miimal -1 1 value of the ormal curvature k(v). W 5

6 Theorem. (Euler s formula) Let v,...,v be a orthoormal basis of T M cosistig of ricial curvature directios, k (),...,k () be the 1-1 corresodig ricial curvatures. The the ormal curvature k(v) i the directio v e T M, N v N = 1, is give by k(y) = S k () <Y,Y > = S k () cos (q ), i i i i where q = arc cos(<v,v >) is the agle betwee v ad v. i i i Proof. Sice (v,...,v ) is a orthoormal basis, the vector v ca be exressed as -1 v = S <v,v > v. i i Makig use of this formula, we obtai -1-1 & * k(v) = <L (v),v> = <L S <v,v > v, S <v,v > v > = 7 i i8 i i = < S <v,v > k () v, S <v,v > v > = S k () <v,v > i i i i i i i The determiat ad trace of the Weigarte ma, that is the roduct ad sum of the ricial curvatures are of articular imortace i differetial geometry. Defiitio. For M a hyersurface, e M, the determiat K() of the Weigarte ma L is called the Gaussia or Gauss-Kroecker curvature of M at , H() = 1/(-1) trace (L ) is called the mea curvature Whe we comute the ricial curvatures ad directios of a hyersurface at a oit we geerally work with a matrix reresetatio of the Weigarte ma. Recall that if V is a liear sace with basis x,...,x ad L:V L V is 1 a liear maig the the matrix reresetatio of V with resect to the basis x,...,x is the x matrix (l ) for which 1 ij L(x ) = S l x,2,...,. i ij j j=1 Whe we have a deal with a arameterized hyersurface r:w l R, it is atural to take the basis r (u),...,r (u) of the taget sace at r(u). Let 1-1 us deote by G = (g ), B = (b ) ad L = (l ) the matrix reresetatios of ij ij ij the first ad secod fudametal forms ad the Weigarte ma resectively, with resect to this basis ( g, b ad l are fuctios o the arameter ij ij ij domai). Comoets of G ad B ca be calculated accordig to the equatios g = < r, r >, ij i j b = < N, r > (cf. Lemma above). ij ij 6

7 The relatioshi betwee the matrices G, B, ad L follows from the followig equalities -1-1 b = < L r, r > = < S l r, r > = S l <r, r > = ij r i j ik k j ik k j k=1 k=1-1 = S l g ik kj k=1 exressig that B = L G. G is the matrix of a ositive defiite biliear fuctio, hece it is ivertible (its determiat is ositive). Multilyig the equatio B = L G with the iverse of G we get the exressio u o -1 L = B G m for the matrix of the Weigarte oerator. Corollary. The Gaussia curvature of a hyersurface is equal to u o det B K = det G m Recall from liear algebra that i order to determie the eigevalues of a liear maig with matrix reresetatio L oe has to fid the roots of the characteristic olyomial (l) = det (L - l I), where I deotes the idetity L matrix. Havig determied the eigevalues of the liear maig, comoets of eigevectors with resect to the fixed basis are obtaied as o-zero solutios of the liear system of equatios L v = l v where l is a ozero eigevalue of L. Further Exercices 5-1. Determie the Weigarte ma for a shere of radius r at oe of its oits Fid the ormal curvature k(v) for each taget directio v, the ricial curvatures ad the ricial curvature directios, ad comute the Gaussia ad mea curvatures of the followig surfaces at the give oit (x /a )+(x /b )+(x /c ) = 1, = (a,0,0) (ellisoid); (x /a )+(x /b )-(x /c ) = 1, = (a,0,0) (oe-sheeted hyerboloid); & 2 2 * x + r x +x - 2 = 1 = (0,3,0) or = (0,1,0) (torus). 7

8 5-3. Suose that the ricial curvatures of a arameterized surface 3 i R vaish. Show that the surface is a art of a lae Fid the Gaussia curvature K:M L R for the followig surfaces x +x -x = 0, x 3 > 0 (coe) (x /a )+(x /b )-(x /c ) = 1 (hyerboloid); (x /a )+(x /b )-x = 0 (ellitic araboloid); (x /a )-(x /b )-x = 0 (hyerbolic araboloid) Let M be a (hyer)surface i R, e M. Show that for each v,w e T M, L (v) x L (w) = K() v x w. 8