### A unified approach to some theorems on homogeneous Riemannian and affine spaces

Rosa Anna Marinosci (1987)

Czechoslovak Mathematical Journal

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Rosa Anna Marinosci (1987)

Czechoslovak Mathematical Journal

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Cristiana Bondioli (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $M$ be a Riemannian manifold, which possesses a transitive Lie group $G$ of isometries. We suppose that $G$, and therefore $M$, are compact and connected. We characterize the Sobolev spaces ${W}_{p}^{1}\left(M\right)$ $\left(1<p<+\mathrm{\infty}\right)$ by means of the action of $G$ on $M$. This characterization allows us to prove a regularity result for the solution of a second order differential equation on $M$ by global techniques.

Dimitri V. Alekseevsky, Andreas Arvanitoyeorgos (2002)

Commentationes Mathematicae Universitatis Carolinae

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A geodesic of a homogeneous Riemannian manifold $(M=G/K,g)$ is called homogeneous if it is an orbit of an one-parameter subgroup of $G$. In the case when $M=G/H$ is a naturally reductive space, that is the $G$-invariant metric $g$ is defined by some non degenerate biinvariant symmetric bilinear form $B$, all geodesics of $M$ are homogeneous. We consider the case when $M=G/K$ is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group $G$, and we give a simple necessary condition that $M$ admits a non-naturally...

Jacek Gancarzewicz, Modesto R. Salgado (1991)

Czechoslovak Mathematical Journal

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