# Category: Differential geometry of surfaces

Differential geometry of surfaces
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 ext
Asymptotic curve
In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, althoug
First fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3
Riemannian connection on a surface
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the e
Third fundamental form
In differential geometry, the third fundamental form is a surface metric denoted by . Unlike the second fundamental form, it is independent of the surface normal.
First Hurwitz triplet
In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera
Tangent developable
In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out
Translation surface (differential geometry)
In differential geometry a translation surface is a surface that is generated by translations: * For two space curves with a common point , the curve is shifted such that point is moving on . By this
Angenent torus
In differential geometry, the Angenent torus is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains self-similar as it evolves under the mean curv
Parabolic line
In differential geometry, a smooth surface in three dimensions has a parabolic point when the Gaussian curvature is zero. Typically such points lie on a curve called the parabolic linewhich separates
Liberman's lemma
Liberman's lemma is a theorem used in studying intrinsic geometry of convex surface.It is named after Joseph Liberman.
Constant-mean-curvature surface
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case.
Constant curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining
Darboux frame
In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux fram
Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map
Dupin indicatrix
In differential geometry, the Dupin indicatrix is a method for characterising the local shape of a surface. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the i
Bertrand–Diguet–Puiseux theorem
In the mathematical study of the differential geometry of surfaces, the Bertrand–Diguet–Puiseux theorem expresses the Gaussian curvature of a surface in terms of the circumference of a geodesic circle
Mean curvature
In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambie
Lidinoid
In differential geometry, the lidinoid is a triply periodic minimal surface. The name comes from its Swedish discoverer Sven Lidin (who called it the HG surface). It has many similarities to the gyroi
Systoles of surfaces
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its syst
Macbeath surface
In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface. The automorphism group of the Macbea
Minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because the
Peano surface
In mathematics, the Peano surface is the graph of the two-variable function It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and min
Hurwitz quaternion order
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientat
Umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions ar
Weingarten equations
The Weingarten equations give the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of a point on the surface. These formulas
Clairaut's relation (differential geometry)
In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which
Second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by (read
Ruled surface
In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a c
Developable surface
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can
Wente torus
In differential geometry, a Wente torus is an immersed torus in of constant mean curvature, discovered by Henry C. Wente. It is a counterexample to the conjecture of Heinz Hopf that every closed, comp
Pu's inequality
In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves co
Simons' formula
In the mathematical field of differential geometry, the Simons formula (also known as the Simons identity, and in some variants as the Simons inequality) is a fundamental equation in the study of mini
Carathéodory conjecture
In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 192
Gauss–Codazzi equations
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental form
Theorema Egregium
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that p
Line of greatest slope
In topography, the line of greatest slope is a curve following the steepest slope. In mountain biking and skiing, the line of greatest slope is sometimes called the fall line.
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gaussian radius of curvatur
Euler's theorem (differential geometry)
In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated pr
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary
Filling area conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introduci
Loewner's torus inequality
In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.
Ridge (differential geometry)
In differential geometry, a smooth surface in three dimensions has a ridge point when a line of curvature has a local maximum or minimum of principal curvature. The set of ridge points form curves on