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Abstract
We present some non-vanishing dual Stiefel-Whitney classes of the Grassmann manifolds O(n)/O(4) × O(n − 4) for n = 2 s + 2 and n = 2 s + 3 (s ≧ 3), providing a supplement to results of Hiller, Stong, and Oproiu. Some applications are also mentioned.
Summary
We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G F (n,N) in the Lie group G F (N) defined naturally makes G F (n,N) a totally geodesic submanifold; (2) The imbedding S 7→SO(8) defined by octonians makes S 7a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G F (N) in the sphere ScN^2-1(√N) of gl(N,F)is minimal. Therefore the natural imbedding G F (N)→gl(N,F )is formed by the eigenfunctions of the Laplacian on G F (N).
Abstract
We prove new results about the vector field problem on the real flag manifolds O(n)/O(1) x O(1) x O(n - 2). For some infinite families of these manifolds, we completely solve the vector field problem.
. Mathematical Research Letters , 5 : 743 – 758 , 1998 . [30] P . Sankaran and S . Sarkar . Degrees of maps between Grassmann manifolds . Osaka J. Math ., 46 ( 4 ): 1143 – 1161 , 2009 . [31] E. H . Spanier . Algebraic Topology . McGraw-Hill, Inc
