study curvature-invariant surfaces 2 in S0(n+2)/S0(2) x SO(n). Next, we shall give a classification of totally geodesic surfaces. Finally we shall show that every ...
B A N G - Y E N C H E N 1 A N D H U E I - S H Y O N G LUE
DIFFERENTIAL
GEOMETRY
OF
SO(n+2)/SO(2)xSO(n), I
]. I N T R O D U C T I O N
SO(n+2) be the special orthogonal group of order n+2. Then SO(2) x SO(n) can be considered as a Lie subgroup of SO(n+2). With respect to a natural Kaeherian structure, SO (n + 2)/S0 (2) x SO (n) forms an Einstein
Let
Hermitian symmetric space of real dimension 2n. It is well-known that this Einstein Hermitian symmetric space is complex analytic isometric to the complex sphere Qn, which is the simplest nonsingular algebraic variety next to complex projective space. In this series of papers, we shall study the differential geometry of this Hermitian symmetric space. In this first part of this series, we shall first study curvature-invariant surfaces 2 in S0(n+2)/S0(2) x SO(n). Next, we shall give a classification of totally geodesic surfaces. Finally we shall show that every totally geodesic surface of S0(n+2)/S0(2)x SO(n) is either a complex surface, a totally real surface, or a surface of type II (for the definition, see Section 4). Moreover, the only fiat totally geodesic surfaces are totally real surfaces of types VI, VII and VIII. 2. PRELIMINARIES
G=SO(n+2) be the special orthogonal group of order n+2, H=SO(2)x SO(n) a closed Lie subgroup of G and In the n by n identity Let
metrix. On G we define an involutive automorphism ~ by
a(A) = SAS-l' where S = ( ' / 2
InO)"
Then, with respect to the canonical connection of (G, H, a), the homogeneous space G/H is an affine symmetric space. Let o (n + 2) denote the Lie algebra of SO (n + 2). We put = 0(2) +
o(n) =
0
B~o(n),
0
2~R)
1 Work done under partial support by NSF Grant GP-36684. 2 Surfaces are of real 2-dimensional.
Geometriae Dedicata 4 (1975) 253-261. All Rights Reserved Copyright © 1975 by Reidel Publishing Company, Dordrecht-Holland
254
BANG-YEN CHEN AND HUEI-SHYONG LUE
and lrt=
5
0
r/
~, ~ are column vectors in R").
Then o(n+2)=~)+m is the canonical decomposition of o(n+2). In the following we shall identify (~, r/)~R" + R" with the matrix
(i ° 3 0
~1
•
On m x m we define an inner product g by (2.1)
g((~,r/),(~',n')) = ,
where +
ab (~, 2>
= (b z + 1) QI, ~'). From (3.7) and the assumption I#1= 12'1, we have (3.11)
a (2, #') + bl#l 2 = 0
(3.4) and (3.11) imply that (3.12)
2 (a s - b') I#l 2 = a QI, 2').
(3.3), (3.4) and (3A1) imply that (3.13)
b (2a ~ -
2b z + 1) I~12 -- - 2 a 2 .
Substituting (3.11), (3.12) and (3.13) into (3.8) we find that either b = 0 or a2-b2=¼. Case (a). b=O. In this case, (3.11) implies (3.14)
a (~, ~') = 0.
Since a ¢ 0 by virtue of (3.9), we have (3.15)
(~, ~'> = 0.
SO(n +2)/S0(2) x SO(n), I
257
(3.3) and (3.15) imply (3.16)
= 0.
(3.5), (3.9), (3.10) and (3.16) imply (3.17)
a = +½,
Iwl = 21#1, 1
