Isoparametric Function

	Isoparametric Function In differential geometry, an isoparametric function is a function on a Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ... whose level surfaces are parallel and of constant mean curvatures. They were introduced by . See also * Isoparametric manifold References * Riemannian geometry {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Riemannian Manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Isoparametric Manifold In Riemannian geometry, an isoparametric manifold is a type of (immersed) submanifold of Euclidean space whose normal bundle is flat and whose principal curvatures are constant along any parallel normal vector field. The set of isoparametric manifolds is stable under the mean curvature flow. Examples A straight line in the plane is an obvious example of isoparametric manifold. Any affine subspace of the Euclidean n-dimensional space is also an example since the principal curvatures of any shape operator are zero. Another simplest example of an isoparametric manifold is a sphere in Euclidean space. Another example is as follows. Suppose that ''G'' is a Lie group and ''G''/''H'' is a symmetric space with canonical decomposition :\mathbf = \mathbf\oplus\mathbf of the Lie algebra g of ''G'' into a direct sum (orthogonal with respect to the Killing form) of the Lie algebra h or ''H'' with a complementary subspace p. Then a principal orbit of the adjoint representation of ''H'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]