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Geodesic orbit metrics in compact homogeneous manifolds with equivalent isotropy submodules. (English) Zbl 1404.53054

The author studies the classification of GO-metrics on compact homogeneous manifolds \(M\cong G/H\).
Using a result based on a proposition by D. Alekseevsky and A. Arvanitoyeorgos [Trans. Am. Math. Soc. 359, No. 8, 3769–3789 (2007; Zbl 1148.53038)], the problem of finding G-GO metrics on \(M\) is reduced to the algebraic problem of determining metric endomorphisms \(A:\mathfrak{m}\rightarrow \mathfrak{m}\) with the property that for any vector \(X\in \mathfrak{m}\), there exists a vector \(a_X\in \mathfrak{h}\) such that \([a_X+X,AX] = 0\). Here, \(\mathfrak{g},\mathfrak{h}\) are the Lie algebras of \(G,H\) respectively, and \(\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}\) is the reductive decomposition with respect to an \(\mathrm{Ad}\)-invariant inner product in \(\mathfrak{g}\). In particular, \(\mathfrak{m}\cong_{\mathrm{iso}} T_pM\).
Several results are presented to simplify the study of the algebraic formulation of the problem. These are used to study the \(\operatorname{U}(n)\)-GO metrics on the complex Stiefel manifolds \(V_k\mathbb{C}^n\), proving that there exists a unique (up to scalar) one-parameter family \(A_t\) of \(\operatorname{U}(n)\)-GO metrics in \(V_k\mathbb{C}^n\) (see Theorem 1).

MSC:

53C22 Geodesics in global differential geometry
53C30 Differential geometry of homogeneous manifolds
53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 1148.53038
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References:

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