# Gauss map

Last updated

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 N: XS2 such that N(p) is a unit vector orthogonal to X at p, namely the normal vector to X at p.

## Contents

The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator.

Gauss first wrote a draft on the topic in 1825 and published in 1827.

There is also a Gauss map for a link, which computes linking number.

## Generalizations

The Gauss map can be defined for hypersurfaces in Rn as a map from a hypersurface to the unit sphere Sn 1  Rn.

For a general oriented k-submanifold of Rn the Gauss map can also be defined, and its target space is the oriented Grassmannian ${\displaystyle {\tilde {G}}_{k,n}}$, i.e. the set of all oriented k-planes in Rn. In this case a point on the submanifold is mapped to its oriented tangent subspace. One can also map to its oriented normal subspace; these are equivalent as ${\displaystyle {\tilde {G}}_{k,n}\cong {\tilde {G}}_{n-k,n}}$ via orthogonal complement. In Euclidean 3-space, this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as ${\displaystyle {\tilde {G}}_{1,n}\cong S^{n-1}}$), hence this is consistent with the definition above.

Finally, the notion of Gauss map can be generalized to an oriented submanifold X of dimension k in an oriented ambient Riemannian manifold M of dimension n. In that case, the Gauss map then goes from X to the set of tangent k-planes in the tangent bundle TM. The target space for the Gauss map N is a Grassmann bundle built on the tangent bundle TM. In the case where ${\displaystyle M=\mathbf {R} ^{n}}$, the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.

## Total curvature

The area of the image of the Gauss map is called the total curvature and is equivalent to the surface integral of the Gaussian curvature. This is the original interpretation given by Gauss. The Gauss–Bonnet theorem links total curvature of a surface to its topological properties.

${\displaystyle \iint _{R}\pm |N_{u}\times N_{v}|\ du\,dv=\iint _{R}K|X_{u}\times X_{v}|\ du\,dv=\iint _{S}K\ dA}$

## Cusps of the Gauss map

The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, (that is along a parabolic line) the Gauss map will have a fold catastrophe. This fold may contain cusps and these cusps were studied in depth by Thomas Banchoff, Terence Gaffney and Clint McCrory. Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface. Cusps occur when:

1. The surface has a bi-tangent plane
2. A ridge crosses a parabolic line
3. at the closure of the set of inflection points of the asymptotic curves of the surface.

There are two types of cusp: elliptic cusp and hyperbolic cusps.

## Related Research Articles

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.

The Gauss–Bonnet theorem, or Gauss–Bonnet formula, is a relationship between surfaces in differential geometry. It connects the curvature of a surface to its Euler characteristic.

In differential geometry, the Gaussian curvature or Gauss curvatureΚ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point:

In mathematics, the GrassmannianGr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V) is the space of lines through the origin in V, so it is the same as the projective space of one dimension lower than V.

In mathematics, the Chern theorem states that the Euler-Poincaré characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature form.

In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.

In Riemannian geometry, the geodesic curvature of a curve measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold , the geodesic curvature is just the usual curvature of . However, when the curve is restricted to lie on a submanifold of , geodesic curvature refers to the curvature of in and it is different in general from the curvature of in the ambient manifold . The (ambient) curvature of depends on two factors: the curvature of the submanifold in the direction of , which depends only on the direction of the curve, and the curvature of seen in , which is a second order quantity. The relation between these is . In particular geodesics on have zero geodesic curvature, so that , which explains why they appear to be curved in ambient space whenever the submanifold is.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.

In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion.

In differential geometry, 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.

A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves. In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting".

Thomas Francis Banchoff is an American mathematician specializing in geometry. He is a professor at Brown University, where he has taught since 1967. He is known for his research in differential geometry in three and four dimensions, for his efforts to develop methods of computer graphics in the early 1990s, and most recently for his pioneering work in methods of undergraduate education utilizing online resources.

In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of a Riemannian or pseudo-Riemannian manifold.

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 are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction. The name "umbilic" comes from the Latin umbilicus - navel.

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, 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 early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.

In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: EB, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map

## References

• Gauss, K. F., Disquisitiones generales circa superficies curvas (1827)
• Gauss, K. F., General investigations of curved surfaces, English translation. Hewlett, New York: Raven Press (1965).
• Banchoff, T., Gaffney T., McCrory C., Cusps of Gauss Mappings, (1982) Research Notes in Mathematics 55, Pitman, London. online version
• Koenderink, J. J., Solid Shape, MIT Press (1990)
• Weisstein, Eric W. "Gauss Map". MathWorld .
• Thomas Banchoff; Terence Gaffney; Clint McCrory; Daniel Dreibelbis (1982). Cusps of Gauss Mappings. Research Notes in Mathematics. 55. London: Pitman Publisher Ltd. ISBN   0-273-08536-0 . Retrieved 4 March 2016.