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The Gauss formula [6] now asserts that is the Levi-Civita connection for M, and is a symmetric vector-valued form with values in the normal bundle. It is often referred to as the second fundamental form. An immediate corollary is the Gauss equation for the curvature tensor.
A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and ...
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "towards" the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. Mean Curvature may ...
Alternatively, the sectional curvature can be characterized by the circumference of small circles. Let be a two-dimensional plane in .Let () for sufficiently small > denote the image under the exponential map at of the unit circle in , and let () denote the length of ().
When a line of curvature has a local extremum of the same principal curvature then the curve has a ridge point. These ridge points form curves on the surface called ridges. The ridge curves pass through the umbilics. For the star pattern either 3 or 1 ridge line pass through the umbilic, for the monstar and lemon only one ridge passes through. [3]
A simple example of mean curvature flow is given by a family of concentric round hyperspheres in +. The mean curvature of an m {\displaystyle m} -dimensional sphere of radius R {\displaystyle R} is H = m / R {\displaystyle H=m/R} .
An example of a complex region where Gauss–Bonnet theorem can apply. Shows the sign of geodesic curvature. In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.