Gaussian curvature

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In mathematics, the Gaussian curvature of a point on a surface is the product of the principal curvatures, κ₁ and κ₂ of the given point.

Symbolically, the Gaussian curvature Κ is defined as

[ \Kappa = \kappa_1 \kappa_2 \,\!].

It is also given by

[\Kappa = \frac_1, \mathbf_2\rangle}]

where [\nabla_i = \nabla__i}] is the covariant derivative and g is the metric tensor.

At a point p on a regular surface in [\mathbb^3], the Gaussian curvature is also given by

[K(\mathbf) = \det(S(\mathbf))]

where S is the shape operator.

Contents

1 Theorema egregium
2 Gauss–Bonnet theorem
3 References
4 See also

Theorema egregium

Gauss's 1828 excellent theorem or Theorema egregium states that the Gaussian curvature depends only on the first fundamental form (metric tensor) and its derivatives and not on the second fundamental form.

A corollary of this theorem is that the Gaussian curvature is invariant under isometric deformations of the surface. Hence the Gaussian curvature of a surface is an intrinsic property of the surface, and can be determined without reference to the embedding of the surface in space. For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat).Porteous, I. R., Geometric Differentiation. Cambridge University Press, 1994. ISBN 052139063X

Gauss–Bonnet theorem

The Gauss-Bonnet theorem links the integral of Gaussian curvature to the Euler characteristic and provides an important link between local geometric properties and global topological properties.

Gaussian curvature

Theorema egregium

Gauss–Bonnet theorem

References

See also