Surface (topology) in the context of Complex analysis

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.

In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

The set of the points that satisfy some property is often called the locus of a point satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be located or may move.

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an ${\displaystyle n}$-dimensional manifold, or ${\displaystyle n}$-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of ${\displaystyle n}$-dimensional Euclidean space.

One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ₁ and κ₂, at the given point:$${\displaystyle K=\kappa _{1}\kappa _{2}.}$$For example, a sphere of radius r has Gaussian curvature ⁠1/r⁠ everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

Gaussian curvature is an intrinsic measure of curvature, meaning that it could in principle be measured by a 2-dimensional being living entirely within the surface, because it depends only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the Theorema Egregium.

In mathematics, the Klein bottle (/ˈklaɪn/) is an example of a surface with no distinct inside or outside. In other words, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, it is an example of a non-orientable surface, a two-dimensional manifold on which one cannot define a consistent direction perpendicular to the surface (normal vector) that varies continuously over the whole shape.

The Klein bottle is related to other non-orientable surfaces like the Möbius strip, which also has only one side but does have a boundary. In contrast, the Klein bottle is boundaryless, like a sphere or torus, though it cannot be embedded in ordinary three-dimensional space without intersecting itself.

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". It generalizes the concept of curve orientation, which for a plane simple closed curve is defined based on whether the curve interior is to the left or to the right of the curve. A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space.

Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. A generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σ_p) depends on a two-dimensional linear subspace σ_p of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ_p as a tangent plane at p, obtained from geodesics which start at p in the directions of σ_p (in other words, the image of σ_p under the exponential map at p). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

The sectional curvature determines the Riemann curvature tensor completely.

In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).

In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982) as part of his 24 questions, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

In the context of a spatial index, a grid or mesh is a regular tessellation of a manifold or 2-D surface that divides it into a series of contiguous cells, which can then be assigned unique identifiers and used for spatial indexing purposes. A wide variety of such grids have been proposed or are currently in use, including grids based on "square" or "rectangular" cells, triangular grids or meshes, hexagonal grids, and grids based on diamond-shaped cells. A "global grid" is a kind of grid that covers the entire surface of the globe.

In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector. Similarly, a vector at a point on a surface can be broken down the same way.

More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point of N, it can be decomposed into the component tangent to N and the component normal to N.

In mathematics, genus (pl.: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.