Equivariant minimal immersions of compact Riemannian homogeneous spaces into compact Riemannian homogeneous spaces

Harmonic mappings, minimal and totally geodesic immersions of compact Riemannian homogeneous spaces into Grassmann manifolds

“…The existence and construction of minimal immersions and harmonic mappings are interesting and important problems in various situations. In the previous paper [1], we construct harmonic mappings and minimal immersions from compact Riemannian homogeneous spaces into Grassmann manifolds. In this paper, we study different construction of harmonic mappings, minimal and totally geodesic immersions of compact Riemannian homogeneous spaces into Grassmann manifolds (see Theorem A and B).…”

“…Let G be a compact connected Lie group with Lie algebra g and K be a closed subgroup of G with Lie algebra f. Take a bi-invariant Riemannian metric < , > on G and denote also by <, > the induced Ad(G)-invariant inner product on m-ϊ 1 . Thus M=(M n , <, »=G//f is a compact Riemannian homogeneous space.…”

“…So if F were not an isometric (more precisely, homothetic) immersion, then F*=0. This means that V P andF Q are G-invariant (see (2.1) in [1]). This is a contradiction.…”

“…V k -the orthogonal projection of span{p(X) k v 0 ; Zem) to (V Q -\ \-Vk-i) 1 -Proof. We prove this by induction on k. It is clear when k=0.…”

“…Put F as in (2.2), then F is a minimal immersion. Since V 0 = V Sf VΊ = V 2 (/f-isomorphic), this example is not contained in Theorem 3.1, [1].…”