In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, the Gaussian curvature or Gauss curvature of a
surface at a point is the product of the
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 point. They measure how the surface bends b ...
s, and , at the given point:
The Gaussian radius of curvature is the reciprocal of .
For example, a sphere of radius has Gaussian curvature 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
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
or the inside of a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
.
Gaussian curvature is an ''intrinsic'' measure of
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
, depending 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''.
Gaussian curvature is named after
Carl Friedrich Gauss, who published the ''
Theorema egregium'' in 1827.
Informal definition
At any point on a surface, we can find a
normal vector that is at right angles to the surface; planes containing the normal vector are called ''
normal planes''. The intersection of a normal plane and the surface will form a curve called a ''
normal section'' and the curvature of this curve is the ''
normal curvature 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 frame exists at any non- umbilic point of ...
''. For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the
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 point. They measure how the surface bends b ...
s, call these , . The Gaussian curvature is the product of the two principal curvatures .
The sign of the Gaussian curvature can be used to characterise the surface.
*If both principal curvatures are of the same sign: , then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
*If the principal curvatures have different signs: , then the Gaussian curvature is negative and the surface is said to have a hyperbolic or
saddle point. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the
asymptotic curves for that point.
*If one of the principal curvatures is zero: , the Gaussian curvature is zero and the surface is said to have a parabolic point.
Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a
parabolic line.
Relation to geometries
When a surface has a constant zero Gaussian curvature, then it is a
developable surface and the geometry of the surface is
Euclidean geometry.
When a surface has a constant positive Gaussian curvature, then the geometry of the surface is
spherical geometry.
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s and patches of spheres have this geometry, but there exist other examples as well, such as the
football.
When a surface has a constant negative Gaussian curvature, then it is a
pseudospherical surface and the geometry of the surface is
hyperbolic geometry.
Relation to principal curvatures
The two principal curvatures at a given point of a
surface are the
eigenvalues of the
shape operator at the point. They measure how the surface bends by different amounts in different directions at that point. We represent the surface by the
implicit function theorem as the graph of a function, , of two variables, in such a way that the point is a critical point, that is, the gradient of vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at is the determinant of the
Hessian matrix of (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2×2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.
Alternative definitions
It is also given by
where is the
covariant derivative and is the
metric tensor.
At a point on a regular surface in , the Gaussian curvature is also given by
where is the
shape operator.
A useful formula for the Gaussian curvature is
Liouville's equation
:''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).''
: ''For Liouville's equation in quantum mechanics, see Von Neumann equation.''
: ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...
in terms of the Laplacian in
isothermal coordinates.
Total curvature
The
surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a
geodesic triangle
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connectio ...
equals the deviation of the sum of its angles from . The sum of the angles of a triangle on a surface of positive curvature will exceed , while the sum of the angles of a triangle on a surface of negative curvature will be less than . On a surface of zero curvature, such as the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
, the angles will sum to precisely radians.
A more general result is the
Gauss–Bonnet theorem.
Important theorems
''Theorema egregium''
Gauss's ''Theorema egregium'' (Latin: "remarkable theorem") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. In fact, it can be found given the full knowledge of the
first fundamental form and expressed via the first fundamental form and its
partial derivatives of first and second order. Equivalently, the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the
second fundamental form of a surface in can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the ''definition'' of the Gaussian curvature of a surface in certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the
intrinsic metric of the surface without any further reference to the ambient space: it is an
intrinsic invariant. In particular, the Gaussian curvature is invariant under
isometric deformations of the surface.
In contemporary
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, a "surface", viewed abstractly, is a two-dimensional
differentiable manifold. To connect this point of view with the
classical theory of surfaces, such an abstract surface is
embedded into and endowed with the
Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface in . A ''local isometry'' is a
diffeomorphism between open regions of whose restriction to is an isometry onto its image. ''Theorema egregium'' is then stated as follows:
For example, the Gaussian curvature of a
cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). On the other hand, since a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
of radius has constant positive curvature and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar representation of even a small part of a sphere must distort the distances. Therefore, no
cartographic projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
is perfect.
Gauss–Bonnet theorem
The Gauss–Bonnet theorem links the total curvature of a surface to its
Euler characteristic and provides an important link between local geometric properties and global topological properties.
Surfaces of constant curvature
*
Minding's theorem (1839) states that all surfaces with the same constant curvature are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called
developable surfaces. Minding also raised the question of whether a
closed surface with constant positive curvature is necessarily rigid.
*
Liebmann's theorem (1900) answered Minding's question. The only regular (of class ) closed surfaces in with constant positive Gaussian curvature are
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s. If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses
Hilbert's lemma that non-
umbilical points of extreme principal curvature have non-positive Gaussian curvature.
*
Hilbert's theorem (1901) states that there exists no complete analytic (class ) regular surface in of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class immersed in , but breaks down for -surfaces. The
pseudosphere has constant negative Gaussian curvature except at its singular
cusp.
There are other surfaces which have constant positive Gaussian curvature.
Manfredo do Carmo considers surfaces of revolution
where
, and
(an
incomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for
either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative Gaussian curvature, one of which is
pseudosphere.
There are many other possible bounded surfaces with constant Gaussian curvature. Whilst the sphere is rigid and can not be bent using an isometry, if a small region removed, or even a cut along a small segment, then the resulting surface can be bent. Such bending preserves Gaussian curvature so any such bending of a sphere with a region removed will also have constant Gaussian curvature.
Alternative formulas
*Gaussian curvature of a surface in can be expressed as the ratio of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
s of the
second and
first fundamental forms and :
*The (after
Francesco Brioschi) gives Gaussian curvature solely in terms of the first fundamental form:
*For an ''
orthogonal parametrization'' (), Gaussian curvature is:
*For a surface described as graph of a function , Gaussian curvature is:
* For an implicitly defined surface, , the Gaussian curvature can be expressed in terms of the gradient and
Hessian matrix :
* For a surface with metric conformal to the Euclidean one, so and , the Gauss curvature is given by ( being the usual
Laplace operator):
*Gaussian curvature is the limiting difference between the
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out ...
of a ''
geodesic circle A geodesic circle is either "the locus on a surface at a constant geodesic distance from a fixed point" or a curve of constant geodesic curvature.
A geodesic disk is the region on a surface bounded by a geodesic circle.
In contrast with the ord ...
'' and a circle in the plane:
[ Bertrand–Diquet–Puiseux theorem]
*Gaussian curvature is the limiting difference between the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a ''
geodesic disk'' and a disk in the plane:
*Gaussian curvature may be expressed with the ''
Christoffel symbols'':
See also
*
Earth's Gaussian radius of curvature
*
Sectional curvature 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 p ...
*
Mean curvature
*
Gauss map
*
Riemann curvature tensor
*
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 point. They measure how the surface bends b ...
References
Books
*
*
External links
*
{{Carl Friedrich Gauss
Curvature (mathematics)
Differential geometry
Differential geometry of surfaces
Surfaces
Differential topology
Carl Friedrich Gauss