Pseudo-Riemannian manifolds
Pseudo-Riemannian metrics
Basic concepts
The aim of this chapter is to introduce the theory of Pseudo-Riemannian geometry, while Öxing Örst the notations that we will use throughout this thesis. We mean by a manifold M a connected smooth manifold, of a Önite dimension n. We denote by TxM the tangent space at the point x 2 M; and by T xM the cotangent space at x. TM denotes the tangent bundle of M and T M the cotangent bundle of M. We consider (M) the space of all tangent vector Öelds of M. For the notation, a vector Öeld will be denoted by capital letters X; Y; Z; :: and the tangent vector at a given point x of the manifold by Xx; Yx; ::: or u; v; :::. Denote by F(M) the commutative ring of all smooth real-valued functions on M. It is a commutative ring for the two operations sum and product of functions.A pseudo-Riemannian metric tensor g on a manifold M is a symmetric nondegenerate (0; 2) tensor Öeld on M of constant index, i.e., g assigns to each point x 2 M a scalar product gx on the tangent space TxM and the index s (the number of negative eigenvalues of gx in an orthonormal basis), is the same for all x 2 M. We can also say that g has signature (s; q), where q = dim M s:
A pseudo-Riemannian manifold M is a smooth manifold equipped with a pseudo-Riemannian metric tensor g. The common value s of index of g on M is called the index of M (0 s dim M) . ñ If s = 0, (M; g) is called a Riemannian manifold. In this case, each gx is a positive deÖnite scalar product on TxM; called inner product with signature (0; n). ñ If s = 1, (M; g) is called a Lorentz manifold and the corresponding metric is called Lorentzian with signature (1; n 1). In particular, that is the metric used in general relativity. A pseudo-Riemannian metric on an even-dimensional manifold M is called a neutral metric if its index is equal to 1 2 dim M: Any smooth manifold can always be equipped with a Riemannian metric, but it is not true for the lorantzian case, we can check for example that there is no Lorentzian metrics on the sphere S 2 : A pseudo-Riemannian manifold (resp. metric) is also known as semiRiemannian manifold .
Isometries
DeÖnition 1.1.12 Let (M; g) and (N; h) be pseudo-Riemannian manifolds and f : M ! N be a smooth map. Then f is called an isometry if f is a di§eomorphism that preserves metric tensors such that f h = If f is an isometry, then so is f 1 . Moreover, the composition of two isometries is an isometry. Finally, the identity map on M is an isometry. The set of isometries of a pseudo-Riemannian manifold M is a Lie group with multiplication given by composition, with Önite dimension said the group of isometries and denoted by Isom (M), such as dim Isom (M) n(n+1) 2
Geodesics
DeÖnition 1.3.1 A geodesic in a pseudo-Riemannian manifold M is a curve : I ! M whose velocity vector Öeld 0 is parallel or covariantly constant along this curve, i.e., r 0 0 = 0: So equivalently, with respect to a coordinate system fx 1 ; :::; xng on an open subset U of M; its coordinate functions sati The Hopf-Rinowís theorem gives that, in the Riemannian case, a manifold (M; g) is geodesically complete if and only if itís metric induces a distance which is complete. Specially, all compact Riemannian manifold is geodesically complete, it is not true in the Lorantzian case. The isometry group of a Riemannian manifold is always compact and complete, but for the Lorentzian manifolds this is not always the case. (see [24], [18] )
Curvature
Riemann curvature tensor
DeÖnition 1.4.1 On a pseudo-Riemannian manifold (M; g) with Levi-Civita connection r, we deÖne the (1; 3) tensor Öeld R from (M) (M) (M) into (M) given in terms of the covariant derivative
Sectionnal curvature
Let (M; g) be a pseudo-Riemannian manifold equipped with the Levi-Civita connection r. Let x a point of M and P a 2-dimensional linear subspace of the tangent space TxM called a plane section. For a given basis fu; vg of the plane section P, we deÖne a real number which is called the sectional curvature and it is independant of the choice of the basis fu; vg of P. If the sectional curvature K of a pseudo-Riemannian manifold M vanishes identically then its curvature tensor R is zero at every point and it is said to be áat. For each index s, the pseudo-Euclidean nspace E n s is áat. In fact, the Christo§el symbols all vanish for a natural coordinate system. Then, the curvature tensor of E n s vanishes identically. With respect to any local orthonormal basis feigi=1;::;n of the tangent bundle, we have that the sectional curvature tensor is given locally.
Ricci curvature tensor
The Ricci curvature (or Ricci tensor) of M; is the symmetric (0; 2) tensor deÖned for all X; Y 2 (M) by Ric(X; Y ) = tracefZ 7! R(X; Z)Y g, it is also the contraction of the curvature tensor .
Killing vectors
DeÖnition 1.5.2 On a pseudo-Riemannian manifold (M; g); a Killing vector Öeld X is a vector Öeld on M; for which the Lie derivative of the metric tensor g vanishes, i.e., LXg = 0: If the one-parameter (local) group of di§eomorphisms ‘t , generated by X is an isometry (on its domain), then t (g) = g, so under the áow of a Killing vector Öeld X, the metric tensor does not change; thus a Killing vector Öeld is an inÖnitesimal isometry.
Chapter 2
Generalized symmetric spaces In this chapter, we start by introducing, some fundamental concepts of homogeneous spaces theory, namely pseudo-Riemannian homogeneous and reductive spaces, then symmetric spaces next we recall the deÖnition of generalized symmetric spaces and Önally we present their classiÖcation in dimension 4 due to Cerny and Kowalski [15]. Let (M; g) be a pseudo-Riemannian manifold. If there is a connected Lie group G Isom(M) which acts transitively on (M; g) as a group of isometries, then (M; g) is called a homogeneous pseudo-Riemannian manifold. Let x 2 M be a Öxed point. If we denote by H the isotropy group at x, then M can be identiÖed with the homogeneous space G=H. In general, there may exist more than one such transitive isometry group G Isom(M) and more presentations of M as a homogeneous space. For any Öxed choice M = G=H, G acts e§ectively on G=H from the left. The pseudo-Riemannian metric g on M can be considered as a Ginvariant metric on G=H. The pair (G=H; g) is then called a pseudo-Riemannian homogeneous space. If the metric g is positive deÖnite so Riemannian, then (G=H; g) is always a reductive homogeneous space such as if we denote by g and h the Lie algebras of G and H, respectively, and we consider the adjoint representation Ad : H g ! g of H on g, there exists a direct sum decomposition (reductive decomposition) of the form g = m + h, where m g is a vector subspace Ad(H)invariant i.e., Ad(H)(m) m. Symmetric spaces are a special class of homogeneous spaces [28], they have been introduced in the late of 1920ís by E. Cartan. In particular, Riemannian symmetric spaces were classiÖed by E. Cartan, but that classiÖcation is much more complicated, in the pseudo-Riemannian case.
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Table des matières
Pseudo-Riemannian manifolds
1.1 Pseudo-Riemannian metrics
1.1.1 Basic concepts
1.1.2 Examples of pseudo-Riemannian manifolds
1.1.3 Isometries
1.2 Levi-Civita connection
1.3 Geodesics
1.4 Curvature
1.4.1 Riemann curvature tensor
1.4.2 Sectionnal curvature
1.4.3 Ricci curvature tensor
1.4.4 Scalar curvature
1.5 Killing vectors
1.5.1 Lie derivative
1.5.2 Killing vectors
2 Generalized symmetric spaces
2.1 Homogeneous pseudo-Riemannian spaces
2.2 Symmetric pseudo-Riemannian spaces
2.3 Generalized symmetric spaces
2.3.1 Generalized symmetric pseudo-Riemannian spaces and canonical connection
2.3.2 ClassiÖcation of four dimensional Generalized symmetric pseudo-Riemannian spaces
3 Curvature properties of four-dimensional Generalized symmetric spaces
3.1 Generalized symmetric spaces of Type A
3.1.1 Levi-Civita connection
3.1.2 Curvature tensor
3.1.3 Ricci curvature tensor
3.2 Generalized symmetric spaces of Type B
3.2.1 Levi-Civita connection
3.2.2 Curvature tensor
3.2.3 Ricci curvature tensor
3.3 Generalized symmetric spaces of Type C
3.3.1 Levi-Civita connection
3.3.2 Curvature tensor
3.3.3 Ricci curvature tensor
3.4 Generalized symmetric spaces of Type D
3.4.1 Levi-Civita connection
3.4.2 Curvature tensor
3.4.3 Ricci curvature tensor
3.5 Geodesics on the four-dimensional Generalized Symmetric Spaces of type C
4 Caracterization of Killing vector Öelds on four-dimensional Generalized Symmetric Spaces 4.1 Killing vectors of Type A
4.2 Killing vectors of Type B
4.3 Explicit calculus of Killing vectors of Type C
4.4 Killing vectors of Type D
5 Ricci Solitons
5.1 Ricci Flow
5.2 Ricci solitons
5.2.1 Examples of Ricci solitons
5.2.2 Some properties and results on Ricci Soliton
5.3 Ricci Soliton on four-dimensional pseudo-Riemannian Generalized symmetric spaces
5.3.1 TypeB
5.3.2 Type C
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