topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. Low-dimensional manifolds are classified...

The Classification of Surfaces Lecture Notes by Z.Fiedorowicz History and Art of Surfaces and their Mathematical Models 2-manifolds at the Manifold Atlas...

mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable...

of Platonic solids Classification theorems of surfaces Classification of two-dimensional closed manifolds – Two-dimensional manifoldPages displaying short...

smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and...

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

Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable...

that require only a smooth structure on a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can...

and Fréchet manifolds, in particular manifolds of mappings are infinite dimensional differentiable manifolds. For a Ck manifold M, the set of real-valued...

it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were first given this...

dimensional manifolds. Indeed the assumption was proven for manifolds of dimension ≤ 3 {\displaystyle \leq 3} and for differentiable manifolds but it was...

Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions)...

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner...

Kähler manifolds are symplectic, in Riemannian geometry where complex manifolds provide examples of exotic metric structures such as Calabi–Yau manifolds and...

lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even...

variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics. Since Kähler manifolds are equipped with...

structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles...

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

are versions of the surgery exact sequence depending on the category of manifolds we work with: smooth (DIFF), PL, or topological manifolds and whether...

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

the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic...

physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It is used in...

of two cubes sharing only a single vertex). For polyhedra defined in these ways, the classification of manifolds implies that the topological type of...

There are no known examples of compact quaternion-Kähler manifolds that are not locally symmetric. (Again, hyperkähler manifolds are excluded from the discussion...

several ways). The complete list of such manifolds is given in the article on spherical 3-manifolds. Under Ricci flow, manifolds with this geometry collapse...

Lorentzian manifold. It is most often applied in studying exact solutions of Einstein's field equations, but strictly speaking the classification is a theorem...

timeline of manifolds, one of the major geometric concepts of mathematics. For further background see history of manifolds and varieties. Manifolds in contemporary...

below). Banach manifolds are one possibility of extending manifolds to infinite dimensions. A further generalisation is to Fréchet manifolds, replacing Banach...

constructing Calabi–Yau manifolds with cylindrical ends, resulting in tens of thousands of diffeomorphism types of new examples. These manifolds are important in...

is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different...