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Examples of closed surfaces include the sphere, the torus and the Klein bottle. Examples of non-closed surfaces include an open disk (which is a sphere with a puncture), an open cylinder (which is a sphere with two punctures), and the Möbius strip. A surface embedded in three-dimensional space is
The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all n-spheres, are specified as subspaces of a Euclidean space and inherit a ...
The mapping class groups of surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of above), since they act on Teichmüller space and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface.
A torus is an orientable surface The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. The Roman surface is non-orientable.. In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". [1]
Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface.
So there are 19-dimensional families of complex analytic K3 surfaces with an elliptic fibration, and 18-dimensional moduli spaces of projective K3 surfaces with an elliptic fibration. Example: Every smooth quartic surface X in that contains a line L has an elliptic fibration , given by projecting away from L. The moduli space of all smooth ...
Another example of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces. [ 8 ] [ 9 ] Action of the mapping class group and relation to moduli space
Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue surfaces; Zariski surfaces, surfaces in finite characteristic that admit a purely inseparable dominant rational map from the projective plane