WebNov 26, 2014 · A submanifold Σ of a Riemannian manifold M is said to be totally geodesic if every geodesic in Σ is also a geodesic in M.The existence and classification of totally geodesic submanifolds are fundamental problems in submanifold geometry. The existence problem is closely related to curvature, as the following result by Élie Cartan [] shows: Let … WebJun 1, 1975 · The geometry of such Riemannian submersions has been extensively discussed in [37]. The requirement for the Riemannian submersion to have totally geodesic fibers would be ideal for modelling a ...
Some manifold which is not totally geodesic in a compact manifold
WebAbstract. Let Γ Γ be a lattice in SO0(n,1) S O 0 ( n, 1). We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least 2 2, then Γ Γ is arithmetic. This answers a question of Reid for hyperbolic n n -manifolds and, independently, McMullen for hyperbolic 3 3 -manifolds. WebOct 12, 2024 · So that gives you the first totally geodesic hypersurface in some finite cover of the given thing. And since the hypersurface is the same type of lattice, there is a finite cover of it that has a totally geodesic hypersurface by the same argument. There is a single finite cover of the original thing doing both at once. gold pinboard
(PDF) Riemannian submersions with totally geodesic fibers
WebMar 1, 2024 · For \(n \ge 2\), we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds.As in the real hyperbolic case, our primary result is a superrigidity theorem for … WebJan 17, 2013 · Let M be the flat cylinder R × S 1 ⊂ R × C and N = { ( t, e i t) t ∈ R }, which is a geodesic (hence a complete totally geodesic submanifold of M) minimizing between any two points of N (among the geodesics of N ). But the minimizing geodesic in M between the points ( 0, 1) and ( 2 π, 1) is the segment { ( s, 1) s ∈ [ 0, 2 π ... WebFeb 22, 2024 · We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic … gold pilot wings - tate mcrae