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63 relations: Area, Asymptotic curve, Bertrand–Diguet–Puiseux theorem, Carl Friedrich Gauss, Christoffel symbols, Circumference, Covariant derivative, Curvature, Cusp (singularity), Cylinder, Darboux frame, Determinant, Developable surface, Diffeomorphism, Differentiable manifold, Differential geometry, Differential geometry of surfaces, Eigenvalues and eigenvectors, Embedding, Euclidean geometry, Euler characteristic, Ferdinand Minding, First fundamental form, Gauss map, Gauss–Bonnet theorem, Geodesic, Heinrich Liebmann, Hessian matrix, Hilbert's lemma, Hilbert's theorem (differential geometry), Hyperbolic geometry, Hyperboloid, Implicit function theorem, Intrinsic and extrinsic properties, Intrinsic metric, Invariant (mathematics), Isometry (Riemannian geometry), Isothermal coordinates, Laplace operator, Liouville's equation, Map projection, Mean curvature, Metric tensor, Normal (geometry), Normal plane (geometry), Orthogonal coordinates, Partial derivative, Principal curvature, Pseudosphere, Riemann curvature tensor, ..., Riemannian manifold, Saddle point, Second fundamental form, Sectional curvature, Sphere, Spherical geometry, Surface (topology), Surface integral, Theorema Egregium, Torus, Triangle, Two-dimensional space, Umbilical point. Expand index (13 more) »
Area
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.
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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).
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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, or the area of a geodesic disc.
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Carl Friedrich Gauss
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.
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Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.
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Circumference
In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it.
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Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.
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Curvature
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
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Cusp (singularity)
In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward.
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Cylinder
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.
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Darboux frame
In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Developable surface
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature.
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Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
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Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
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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.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
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Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
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Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
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Ferdinand Minding
Ferdinand Minding (Фердинанд Готлибович Миндинг; &ndash) was a German-Russian mathematician known for his contributions to differential geometry.
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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.
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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.
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Gauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).
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Geodesic
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
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Heinrich Liebmann
Karl Otto Heinrich Liebmann (* 22. October 1874 in Strasbourg; † 12. June 1939 in Munich-Solln) was a German mathematician and geometer.
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Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.
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Hilbert's lemma
Hilbert's lemma was proposed at the end of the 19th century by mathematician David Hilbert.
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Hilbert's theorem (differential geometry)
In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative gaussian curvature K immersed in \mathbb^.
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Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.
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Hyperboloid
In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes.
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Implicit function theorem
In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables.
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Intrinsic and extrinsic properties
An intrinsic property is a property of a system or of a material itself or within.
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Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space.
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Invariant (mathematics)
In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.
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Isometry (Riemannian geometry)
In mathematics, an isometry of a manifold is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points.
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Isothermal coordinates
In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric.
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Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
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Liouville's equation
In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor of a metric on a surface of constant Gaussian curvature: where is the flat Laplace operator.
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Map projection
A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane.
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Mean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
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Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
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Normal (geometry)
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.
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Normal plane (geometry)
A normal plane is any plane containing the normal vector of a surface at a particular point.
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Orthogonal coordinates
In mathematics, orthogonal coordinates are defined as a set of d coordinates q.
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Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
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Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point.
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Pseudosphere
In geometry, the term pseudosphere is used to describe various surfaces with constant negative Gaussian curvature.
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Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes.
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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 \mathrm (read "two").
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Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.
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Sphere
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
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Spherical geometry
Spherical geometry is the geometry of the two-dimensional surface of a sphere.
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Surface (topology)
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.
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Surface integral
In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.
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Theorema Egregium
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Triangle
A triangle is a polygon with three edges and three vertices.
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Two-dimensional space
Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).
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Umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical.
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Redirects here:
Concave curvature, Convex curvature, Gauss curvature, Liebmann's theorem.
References
[1] https://en.wikipedia.org/wiki/Gaussian_curvature
