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The term double torus is occasionally used to denote a genus 2 surface. [4] [5] A non-orientable surface of genus two is the Klein bottle. The Bolza surface is the most symmetric Riemann surface of genus 2, in the sense that it has the largest possible conformal automorphism group. [6] Representations of genus 2 surfaces
The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected. It is equal to the number of handles on it. For instance: A ball has genus 0. A solid torus D 2 × S 1 has genus 1.
It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S 1 in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space.
There is a 2-1 covering map from the torus to the Klein bottle, because two copies of the fundamental region of the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. The universal cover of both the torus and the Klein bottle is the plane R 2.
It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of g tori is 2 − 2g.
g-holed torus (g is the genus) 2 − 2g: ... For instance, a figure-eight shape has more holes than a circle , and a 2-torus (a 2-dimensional ...
The mapping torus M g of a homeomorphism g of a surface S is the 3-manifold obtained from S × [0,1] by gluing S × {0} to S × {1} using g. If S has genus at least two, the geometric structure of M g is related to the type of g in the classification as follows: If g is periodic, then M g has an H 2 × R structure;
There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and ...