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We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M , g) {(M,g)}. For example, if a convex set in (M , g) {(M,g)} is bounded by a smooth hypersurface, then it is strictly convex. [ABSTRACT FROM AUTHOR]

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces. [ABSTRACT FROM AUTHOR]

We prove that singular Riemannian foliations in Euclidean spheres can be defined by polynomial equations. [ABSTRACT FROM AUTHOR]

We investigate orthogonal representations of compact Lie groups from the point of view of their quotient spaces, considered as metric spaces. We study metric spaces which are simultaneously quotients of different representations and investigate properties of the corresponding representations. We obtain some structural results and apply them to study irreducible representations of small copolarity. As an important tool, we classify all irreducible representations of connected groups with cohomogeneity four or five. [ABSTRACT FROM AUTHOR]