Removability of singularities of harmonic maps into pseudo-riemannian manifolds
نویسنده
چکیده
We consider harmonic maps into pseudo-Riemannian manifolds. We show the removability of isolated singularities for continuous maps, i.e. that any continuous map from an open subset of R into a pseudoRiemannian manifold which is two times continuously differentiable and harmonic everywhere outside an isolated point is actually smooth harmonic everywhere. Introduction Given n ∈ N and two nonnegative integers p and q such that p + q = n, a pseudo-Riemannian manifold (N , h) of dimension n and of signature (p, q) is a smooth n-dimensional manifold N equipped with a pseudo-Riemannian metric h, i.e. a section of T ∗N ⊙ T ∗N (where ⊙ is the symmetrised tensor product), such that ∀M ∈ N , hM is a non degenerate bilinear form of signature (p, q). Any local chart φ : N ⊃ U −→ V ⊂ R allows us to use local coordinates (y, · · · , y) ∈ V : we then denote by hij(y) := hφ−1(y) ( ∂ ∂yi , ∂ ∂yj ) . We say that (N , h) is of class Ck if and only if hij is Ck. We define the Christoffel symbol by Γjk(y) := 1 2 h(y) ( ∂hlk ∂yj (y) + ∂hjl ∂yk (y)− ∂hjk ∂yl (y) ) ,
منابع مشابه
Harmonic Maps with Prescribed Singularities into Hadamard Manifolds
Let M a Riemannian manifold of dimension m ≥ 3, let Σ be a closed smooth submanifold of M of co-dimension at least 2, and let H be a Hadamard manifold with pinched sectional curvatures. We prove the existence and uniqueness of harmonic maps φ : M \ Σ → H with prescribed singularities along Σ. When M = R, and H = H C , the complex hyperbolic space, this result has applications to the problem of ...
متن کاملCommutative curvature operators over four-dimensional generalized symmetric spaces
Commutative properties of four-dimensional generalized symmetric pseudo-Riemannian manifolds were considered. Specially, in this paper, we studied Skew-Tsankov and Jacobi-Tsankov conditions in 4-dimensional pseudo-Riemannian generalized symmetric manifolds.
متن کاملOn the Dirichlet Problem for Harmonic Maps with Prescribed Singularities
Let (M, g) be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into (M, g) with prescribed singularities along a closed submanifold of the domain. This generalizes our previous work where such maps into the hyperbolic plane were constructed. This problem, in the case wh...
متن کاملList of contributions
Harmonic maps between Riemannian manifolds are maps which extremize a certain natural energy functional; they appear in particle physics as nonlinear sigma models. Their infinitesimal deformations are called Jacobi fields. It is important to know whether the Jacobi fields along the harmonic maps between given Riemannian manifolds are integrable, i.e., arise from genuine variations through harmo...
متن کامل