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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: X → S2 such that N(p) is a unit vector orthogonal to X at p, namely the normal vector to X at p.
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.
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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 , 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 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 ), 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 , the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.
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.
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:
There are two types of cusp: elliptic cusp and hyperbolic cusps.
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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:
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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.
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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.
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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.
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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.
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