In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, dianalytic manifolds are possibly
non-orientable generalizations of
complex analytic manifolds. A dianalytic structure on a manifold is given by an
atlas
An atlas is a collection of maps; it is typically a bundle of world map, maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets.
Atlases have traditio ...
of
charts
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent t ...
such that the
transition maps are either
complex analytic maps or
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
s of complex analytic maps. Every dianalytic manifold is given by the
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of an analytic manifold (possibly non-connected) by a fixed-point-free
involution
Involution may refer to: Mathematics
* Involution (mathematics), a function that is its own inverse
* Involution algebra, a *-algebra: a type of algebraic structure
* Involute, a construction in the differential geometry of curves
* Exponentiati ...
changing the complex structure to its complex conjugate structure. Dianalytic manifolds were introduced by , and dianalytic manifolds of 1 complex dimension are sometimes called
Klein surface
In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves ov ...
s.
References
*
Riemann surfaces
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