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[1] Given an open region D in R 2 , let g and h be symmetric 2-tensors on D , with g additionally required to be positive-definite. If these are smooth and satisfy the Gauss–Codazzi equations, then Bonnet's theorem says that D is covered by open sets which can be smoothly embedded into R 3 with first fundamental form g and second fundamental ...
In the special case of surfaces, this result was proved by Ossian Bonnet in 1855. For a surface, the Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the sectional curvature. Myers' key contribution was therefore to show that a Ricci ...
Pierre Ossian Bonnet (French:; 22 December 1819, Montpellier – 22 June 1892, Paris) was a French mathematician. He made some important contributions to the differential geometry of surfaces, including the Gauss–Bonnet theorem .
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. In the simplest application, the case of a triangle on a plane , the sum of its angles is 180 degrees. [ 1 ]
In classical mechanics, Bonnet's theorem states that if n different force fields each produce the same geometric orbit (say, an ellipse of given dimensions) albeit with different speeds v 1, v 2,...,v n at a given point P, then the same orbit will be followed if the speed at point P equals
The Chern–Gauss–Bonnet theorem can be seen as a special instance in the theory of characteristic classes. The Chern integrand is the Euler class. Since it is a top-dimensional differential form, it is closed. The naturality of the Euler class means that when changing the Riemannian metric, one stays in the same cohomology class. That means ...
The cells in the human body are not outnumbered 10 to 1 by microorganisms. The 10 to 1 ratio was an estimate made in 1972; current estimates put the ratio at either 3 to 1 or 1.3 to 1. [301] The total length of capillaries in the human body is not 100,000 km.
The bonnethead (Sphyrna tiburo), also called a bonnet shark or shovelhead, [3] is a small member of the hammerhead shark genus Sphyrna, and part of the family Sphyrnidae.It is an abundant species in the littoral zone of the North Atlantic and the Gulf of Mexico, is the only shark species known to display sexual dimorphism in the morphology of the head, and is the only shark species known to be ...
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