Prescribed Riemannian Symmetries

Riemannian Symmetries in Flag Manifolds

Flag manifolds are in general not symmetric spaces. But they are provided with a structure of Z2 -symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. The conditions for a metric adapted to the Z2-symmetric structure to be naturally reductive are detailed for the flag manifold SO(5)/SO(2)× SO(2)× SO(1).

Topology of Riemannian Submanifolds with Prescribed Boundary

We prove that a smooth compact immersed submanifold of codimension 2 in R, n ≥ 3, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimenion is too high, or the prescribed ...

Coloring Quasicrystals with Prescribed Symmetries and Frequencies

Symmetries of Flat Rank Two Distributions and Sub-riemannian Structures

Flat sub-Riemannian structures are local approximations — nilpotentizations — of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5. 1. Sub-Riemannian structures A sub-Riemannian geometry is a triple (M,∆, 〈·, ·〉), where M is a smooth manifold, ∆ ⊂ TM is a s...

Manifolds of Riemannian Metrics with Prescribed Scalar Curvature by Arthur E. Fischer and Jerrold

THEOREM 2. Assume J * V 0 . Writing UtQ=(je a o\0 )U&9 J(\ is the disjoint union of closed submanifolds. REMARK. If d i m M = 2 , e^J=^" 8 , and if d i m M = 3 , the hypothesis that 1F*J£0 can be dropped. The proof of Theorem 1 also allows us to conclude that a solution h of the linearized equations DR(g0) • h=0 is tangent to a curve of exact solutions of R(g)=p through a given solution g0, pro...