33 JINWEI ZHOU acts on the left of u =(u ::: u N ) 2 F N Let G(N) be the group acting on the right of F N which preserving the standard inner product

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1 SOOHOW JOURNL OF MTHEMTIS Volume 24, No 4, pp , October 998 THE GEODESIS IN GRSSMNN MNIFOLDS Y JINWEI ZHOU bstract With a suitable basis of Euclidean space, every geodesic in Grassmann manifold can be expressed by (t) = [cos( t)e +sin( t)e n+ ::: cos( nt)e n +sin( nt)e 2n] where 2 n and N;n+ = = n =ifn <2n Introduction Let G F (n N) be the Grassmann manifold formed by all n-subspaces in F N, where F is the set of real numbers, complex numbers or quaternions The manifold G F (n N) is a symmetric space (see [2] or [3]) In this short paper we study the geodesics in Grassmann manifolds We show that every geodesic in Grassmann manifolds lies in a at totally geodesic submanifold which isisometric to a torus In this way, we can write geodesics in Grassmann manifolds explicitly s [4], the results of this paper can be used to study the dierential geometry of Grassmann manifolds s a byproduct, we nd a simple method of the diagonalization of the F -matrices 2 The Geodesics in G F (n N) Let F be the set of real number R, complex numbers or quaternions H For any u 2 F, u is the conjugation of u(u v = v u if u v 2 H) For any 2 F, Received March, 998 MS Subject lassication 5322 Key words Grassmann manifold, geodesic, moving frame 329

2 33 JINWEI ZHOU acts on the left of u =(u ::: u N ) 2 F N Let G(N) be the group acting on the right of F N which preserving the standard inner product (,) on F N If F = R, G(N) =O(N) is the orthogonal group if F =, G(N) =U(N) is the complex unitary group if F = H, G(N) =S p (N) is the symplectic group G(N) G(n)G(N ;n) Let G F (n N) = be the Grassmann manifold formed by all n-subspaces in F N Denote = [e ::: e n ] if 2 G F (n N) is generated by e ::: e n Let e ::: e N be orthonormal frame elds on F N y the method of moving frame, there are local -forms! dened by de = X! e! +! = = ::: N: Restricting the two form = Re( P! i! i ) on G F (n N) denes a Riemannian metric (see []) s is well known, the tangent spacet G F (n N) can be realized as the space of all F valued n (N ; n) matrices Let E i (i = ::: n = n + ::: N) be basis of T G F (n N) dened as follows a curve in G F (n N) Let (t) = [e (t) ::: e n (t)] be omplete e (t) ::: e n (t) to orthonormal frame elds e (t) ::: e n (t) e n+ (t) :::, e N (t) Set a i =( de i dt e ) t= Then the tangent vector of (t) at = () is X = X a i E i : The tanget vector X can also be represented by ann (N ; n) matrix =(a i ) If e ::: e n e n+ ::: e N e e n = is another frame with e e n e n+ e N = D e n+ where e = e () and 2 G(n), D 2 G(N ; n) Then the tangent vector X with this new frame can be represented by t D The tangentspacet G (n N) of complex Grassmann manifold is a complex space of dimension n(n ; n) and G (n N) is a kaehler manifold ut there is no naturally dened quanternion structure on the tangent space T G H (n N) of quaternion Grassmann manifold (in general, t D 6= t D, for 2 H) The following lemma follows from Lemma 22 e N

3 THE GEODESIS IN GRSSMNN MNIFOLDS 33 Lemma 2 For any X 2 T G F (n N), there exists an orthonormal basis e ::: e n ::: e N on F N, such that =[e ::: e n ] and X = nx i E i n+i i= where E i is determined by the frame e ::: e N and 2 n, N ;n+ = = n =if N < 2n Lemma 22 For any F valued n (N ; n) matrix, there are elements g 2 G(n) and g 2 2 G(N ; n) such that g g 2 = g g 2 = N ;n n where n are real numbers if N < 2n if N 2n Proof In the following we assume that N 2n The case of N < 2n can be proved similarly Let f(u) =(u u) = u t u t for any u 2 F n with (u u) = f is a non-negative function and bounded on F n \ S rn;, where r is the real dimension of F ssuming that = f(u ) is the maximum of f for some u 2 F n \ S rn; Then for any u 2 F n \ S rn; with (u u ) =, we have (u u )+(u u )= Then the real part of (u u ) is zero Replace u by u, 2 F, jj =,we can show that (u u )=: Restricting f on fu 2 F n j(u u )=, juj =g, we can get vector u 2 and 2 = f(u 2 ) In this way, we can obtain a G(n)-frame u ::: u n and some non-negative numbers i = f(u i ), n,(u i u j )=ifi 6= j

4 332 JINWEI ZHOU From (u i u j )= i ij,weknow that we canchoose v i 2 F N ;n such that u i = i v i (v i v j )= ij i i = ::: n: omplete v ::: v n to a G(N ;n)-frame v ::: v n ::: v N ;n on F N ;n We have proved that u v = : n Denote g = u u n u n proof of the lemma 2 G(n) andgt 2 = v v N ;n The following proposition can be proved similarly v N ;n 2 G(N ; n) This completes the Proposition 23 Let be a( )-symmetric matrix of order n, thatis t = Then there is a g 2 G(n) such that gg t = where 2 n are real numbers s is well known, the tangentvectors E i n+i (i minfn N ;ng)oft G F (n N) determine a totally geodesic submanifold in G F (n N) which is isometric to n G F ( 2) G F ( 2): ny geodesic with initial tangent vector X = P i E i n+i i 2 R, lies in a at torus of this submanifold Theorem 24 For any geodesic in G F (n N), there is an orthonormal basis e ::: e N on F N such that can be represented by (t) = [cos( t)e +sin( t)e n+ ::: cos( n t)e n +sin( n t)e 2n )] where 2 n, N ;n+ = = n =if N < 2n

5 THE GEODESIS IN GRSSMNN MNIFOLDS 333 s G F ( N) is a projective space, from Theorem 24 we know that the geodesics in projective space are all closed Wong [4] showed that for any, 2 G F (n N), there is an orthonormal basis e ::: e N of F N and numbers ::: n such that and can be written as =[e ::: e n ] =[cos e +sin e n+ ::: cos n e n +sin n e 2n ] where 2 2 n y Theorem 24, the curve dened by (t) = [cos( t)e + sin( t)e n+ ::: cos( n t)e n +sin( n t)e 2n ] is a minimal geodesic joining and Note that, for any real numbers ::: n, if there are integers k ::: k n and real number t such that i t = k i + i Then (t) = [cos( t)e +sin( t)e n+ ::: cos( n t)e n + sin( n t)e 2n ] is also a geodesic through the points, with length qx (ki + i ) 2 : orollary 25 For any, 2 G F (n N), n 2, N 4, therearecountably many geodesics passing through and The geodesics in oriented real Grassmann manifold can be studied similarly The method can also been used to study the geodesics in some homogeneous spaces References [] S S hern, omplex manifolds without potential theory, Springer-Verlag, New York, 979 [2] S Helgason, Dierential geometry, Lie groups, and symmetric spaces, cademic Press, New York, 978 [3] S Kabayashi and K Nomizu, Foundations of dierential geometry, vol 2, Interscience Publishers, New York, 969 [4] Yung-how Wong, Dierential geometry of Grassmann manifolds, Proc Nat cad Sci US, 57 (967), Department of Mathematics, Suzhou University, Suzhou 256, hina

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