Riemannian Homogeneous Manifold Research Articles

Weak solutions of geometric flows associated to integro-differential harmonic maps

The purpose of this note is to prove the existence of global weak solutions to the flow associated to integro-differential harmonic maps into spheres and Riemannian homogeneous manifolds.

On uniqueness of heat flow of harmonic maps

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N,h) that have sufficiently small renormalized energies, provided that N is either a unit sphere Sk−1 or a Riemannian homogeneous manifold. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for t ≥ t0 > 0 and the unique limit property at time infinity. As a corollary, the uniqueness is shown for heat flow of harmonic maps into any compact Riemannian manifold N whose gradients belong to LqtL l x, for q > 2 and l > n satisfying the Serrin’s condition.

Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces

Let \(M=G/H\) be a compact connected isotropy irreducible Riemannian homogeneous manifold, where \(G\) is a compact Lie group (may be, disconnected) acting on \(M\) by isometries. This class includes all compact irreducible Riemannian symmetric spaces and, for example, the tori \(\mathbb{R }^n/\mathbb{Z }^n\) with the natural action on itself extended by the finite group generated by all permutations of the coordinates and inversions in circle factors. We say that \(u\) is a polynomial on \(M\) if it belongs to some \(G\)-invariant finite dimensional subspace \(\mathcal{E }\) of \(L^2(M)\). We compute or estimate from above the averages over the unit sphere \(\mathcal{S }\) in \(\mathcal{E }\) for some metric quantities such as Hausdorff measures of level set and norms in \(L^p(M)\), \(1\le p\le \infty \), where \(M\) is equipped with the invariant probability measure. For example, the averages over \(\mathcal{S }\) of \(\Vert u\Vert _{L^p(M)}\), \(p\ge 2\), are less than \(\sqrt{\frac{p+1}{e}}\) independently of \(M\) and \(\mathcal{E }\).

Homogeneous geodesics in the flag manifold [formula omitted

A geodesic curve in a Riemannian homogeneous manifold ( M = G / K , g ) is called a homogeneous geodesic if it is an orbit of an one-parameter subgroup of the Lie group G . We investigate G -invariant metrics such that all geodesics are homogeneous for the flag manifold M = SO ( 2 l + 1 ) / U ( l - m ) × SO ( 2 m + 1 ) . By reformulating the problem into a matrix form we show that SO ( 2 ℓ + 1 ) / U ( ℓ - m ) × SO ( 2 m + 1 ) has homogeneous geodesics with respect to any SO ( 2 ℓ + 1 ) -invariant metric if and only if m = 0 . In all other cases this space admits at least one non-homogeneous geodesic. We also give examples of finding homogeneous geodesics in the above flag manifold for special values of l and m .

On the Problem of Optimal Reconstruction

We find lower bounds for linear and Alexandrov's cowidths of Sobolev's classes on Compact Riemannian homogeneous manifolds \(M^{d}\). Using these results we give an explicit solution of the problem of optimal reconstruction of functions from Sobolev's classes \(W^{\gamma}_{p}(M^{d})\) in \(L_{q}(M^{d}), 1 \leq q \leq p \leq \infty\).

Riemannian flag manifolds with homogeneous geodesics

A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group G. We investigate G-invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, its T-root system, to give a simple necessary condition that M admits a non-standard G-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R 2l+2 ) = SO(2l +1)/U(l) of complex structures in R 2l+2 , and the complex projective space CP 2l-1 = Sp(l)/U(1) Sp(l- 1) admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only G-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podesta and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

Classification des variété approximativement kähleriennes homogénes

A Riemannian homogeneous manifold admitting a strict nearly-Kahler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions. The six-dimensional homogeneous spaces, S3 × S3, S6, CP(3) and the flag manifold F(1, 2) have a unique (after a change of scale) nearly-Kahler, invariant structure. For the first one we solve a differential equation on the SU(3)-structure given by Reyes Carrion. For the last two it is obtained by canonical variation of the Kahler structure of the twistor space over a four-dimensional manifold. Finally, from Bar, a nearly-Kahler structure on the sphere S6 corresponds to a constant 3-form on the Riemannian cone R7.

Maximal complexifications of certain Riemannian homogeneous manifolds.

Let M = G/K be a homogeneous Riemannian manifold with dim_C G^C = dim_R G, where G^C denotes the universal complexification of G. Under certain extensibility assumptions on the geodesic flow of M, we give a characterization of the maximal domain of definition in TM for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.

A Canonical Connection for Lie Group Actions

The aim of this paper is to generalize the construction of an Ambrose-Singer connection for Riemannian homogeneous manifolds to the case of cohomogeneity one Riemannian manifolds. Necessary and sufficient conditions are given on a Riemannian manifold (M,g) in order that there exists a Lie group of isometries acting on M with principal orbits of codimension one.

Hyperpolar actions and k-flat homogeneous spaces.

A closed, connected, k-dimensional submanifold of a compact Riemannian manifold M is called a of M if it is flat in the induced metric and totally geodesic. We call M k-flat if every geodesic lies in some of M, and if the group of isometries of M acts transitively on pairs (σ, p) consisting of a σ and a point p ∈ σ. An isometric action on M is called hyperpolar if there exists a connected, closed, flat submanifold of M that meets all orbits orthogonally. We prove that the following three properties for a compact Riemannian manifold M are equivalent: (a) M is a Riemannian homogeneous manifold and admits a cohomogeneity k hyperpolar action with a fixed-point, (b) M is homogeneous, (c) M is a rank k symmetric space. Since 1-flat homogeneous is trivially equivalent to two-point homogeneous, the equivalence of (b) and (c) generalizes the well-known fact that two-point homogeneous spaces are the same as rank 1 symmetric spaces.

Isometric immersion of Riemannian homogeneous manifolds

Theorem. Let M be a compact connected Riemannian homogeneous manifold with irreducibleisotropy action. For an equivariant isametric immersion f of M into a Euclidean space EN {considered as a Euclidean vector space) there exist a finite number of vector subspaces Eo, Eu ・■・ , Er of EN, isometric immersions ft of 1-type of M into Et (7=1, ・・・, r), constant vector v0 in Eo and positiveconstant aX)・・・,ar so that (1) EN=E0+E1-{■・■ +Er (Euclidean direct sum) (2) /=yo+fli/i+-+ar/r.

The Manifolds Covered by a Riemannian Homogeneous Manifold

Introduction. The sphere is known to be the universal covering for complete connected Riemannian manifolds of constant positive curvature. More precisely, if M is an n-dimensional complete connected Riemannian manifold of constant sectional curvature k2 > 0 with k > 0, and if Sn is the sphere of radius k-1 in Euclidean space RI'+', with the induced metric, then there is a covering of M by Sn such that the covering projection is a local isometry. Because of this phenonmenon, the complete connected Riemannian manifolds of constant positive curvature are called the spherical spaceforms. In his thesis, G. Vincent [14] attempted to classify them. Following this line of investigation, we take a compact connected Riemannian homogeneous manifold M and ask which Riemannian manifolds admit M as a Riemannian covering manifold. In Chapter I, this problem is reduced to a problem on discrete subgroups of compact Lie groups: