Cartan already in 1925, and they have been the subject of intensive study since then. For example, they were completely classified by E. Symmetric spaces are then a particular type of homogeneous manifolds satisfying some extra conditions, and they are nowadays relatively well understood. Intuitively, homogeneous manifolds locally look the same, as Riemannian manifolds, around every point. Manifolds of the form G/H, where G is a Lie group and H is a closed subgroup of G, are called coset spaces or homogeneous manifolds. The symmetric Riemannian metric on G/H is the invariant metric under G. Then M is of the form G/H, where H is a closed subgroup of G satisfying some further conditions, which we will not explicitly need here, see for example 1 for more details and further references. Let us thus assume that ( M,g) is a complete symmetric space, and let us denote by G the connected component to the identity of its isometry group. On a first look, this definition does not seem to be very intuitive, but if we assume M to be complete, it can be reformulated in an equivalent, perhaps more understandable way. A Riemannian symmetric manifold is a Riemannian manifold ( M,g) with Riemannian metric g such that for every point p in M there exists an isometry of g reversing every geodesic passing through p and such that p is an isolated fixed point of the isometry 1. The technical definition of a Riemannian symmetric manifold goes as follows. The admissible internal space M 6 is a Riemannian symmetric manifold.In order to get a grasp on the kind of manifolds which satisfy these conditions, let us first consider the following possibility: What the Supergravity analysis of the compactification tells us is that any oriented, spin, six-dimensional compact manifold admitting a parallel spinor is a valid compactification background. Rotation about the Y (vertical) axis of a tridimensional representation of a quadridimensional Calabi-Yau manifold. The purpose of this note is to explore the properties of this class of admissible internal manifolds, showing in particular that they are Calabi-Yau manifolds in a precise sense that we will specify later. There is a class of admissible Heterotic internal manifolds characterized as being six-dimensional, oriented, spin, Riemannian compact manifolds admitting a parallel spinor respect to the Levi-Civita spin connection. In particular, we have finished the second article with the following result: Finally, we prove that after a hyper-KГ¤hler rotation, X can be compactified to the complement of a Kodaira type Id fiber appearing as a singular fiber in a rational elliptic surface ПЂЛ‡:YЛ‡в†’в„™1.In the previous articles ( I, II), we have characterized the simplest class of supersymmetric heterotic compactification backgrounds. In the special case that Y is a rational elliptic surface, or Y=в„™2 we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung-Yau. In complex dimension 2, we prove that if Y is a del Pezzo surface, or a rational elliptic surface, and Dв€€|в€’KY| is a smooth divisor with D2=d, then X=Yв€–D admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. We prove that if X is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat KГ¤hler metric constructed by Tian-Yau.

Calabi yau manifolds