Equigeodesics on some classes of homogeneous spaces

Abstract We study homogeneous curves on some classes of reductive homogeneous spaces G / H which are geodesics with respect to any G-invariant metric on G / H . These curves are called equigeodesics. The spaces we consider are certain Stiefel manifolds V k R n , generalized Wallach spaces and spheres. We give a characterization for algebraic equigeodesics on V 2 R n , V 4 R 6 , SO ( 6 ) / SO ( 3 ) ⋅ SO ( 2 ) , W 6 = U ( 3 ) / U ( 1 ) 3 , W 12 = Sp ( 3 ) / Sp ( 1 ) 3 , S 2 n + 1 ≅ U ( n + 1 ) / U ( n ) and S 4 n + 3 ≅ Sp ( n + 1 ) / Sp ( n ) .