Explore: Manifolds

(e.g. CT scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable

Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global

Shing-Tung Yau (1978), who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex

ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who

Lorentz. After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications

Riemmannian manifold defines a number of associated tensor fields, such as the Riemann curvature tensor. Lorentzian manifolds are pseudo-Riemannian manifolds of

different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding

a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold. In particular, it has been

Haken manifolds and their simple and rigid structure leads quite naturally to algorithms. We will consider only the case of orientable Haken manifolds, as

applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing