Transnormal function

In mathematics , transnormal functions play a role, especially in connection with isoparametric surfaces.

definition

Let it be a Riemannian manifold . A twice continuously differentiable function is called transnormal if there is a twice continuously differentiable function with ${\displaystyle M}$ ${\displaystyle f\colon M\to \mathbb {R} }$ ${\displaystyle b\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle |\nabla f(x)|^{2}=b(f(x))}$

for everyone there. The gradient of and denotes the norm defined by means of the Riemann metric . ${\displaystyle x\in M}$ ${\displaystyle \nabla f}$ ${\displaystyle f}$ ${\displaystyle |\cdot |}$

properties

• The level sets of a transnormal function are parallel surfaces , i. H. they have a constant distance.

• If there is a lower or upper bound , then the level sets of the global minimum and maximum are submanifolds, respectively . (They are called focal manifolds, respectively .) ${\displaystyle f}$ ${\displaystyle V_{-}}$ ${\displaystyle V_{+}}$

• The level sets of regular values are spheres bundle over the focal manifolds.

• Transnormal functions on or are isoparametric . ${\displaystyle S^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

Examples

• Be the standard sphere and the restriction of the homogeneous polynomial on . ( ) Then is a transnormal function. The focal manifold in this case is the union of two spheres of dimensions and . ${\displaystyle M=S^{n}\subset \mathbb {R} ^{n+1}}$ ${\displaystyle f}$ ${\displaystyle x_{1}^{2}+\ldots +x_{r}^{2}-x_{r+1}^{2}-\ldots -x_{n+1}^{2}}$ ${\displaystyle S^{n}}$ ${\displaystyle 1<r<n+1}$ ${\displaystyle f^{2}}$ ${\displaystyle V_{+}}$ ${\displaystyle r-1}$ ${\displaystyle n-r-1}$
• Be also the standard sphere and a point is the distance between and the North Pole (on the sphere, in other words, the angle at zero point between and the North Pole). Then define a transnormal function whose focal manifolds are the North Pole and the South Pole. ${\displaystyle M=S^{n}}$ ${\displaystyle p\in S^{n}}$ ${\displaystyle \theta (p)}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle f(p)=\cos \theta (p)}$ ${\displaystyle f\colon S^{n}\to \left[-1,1\right]}$
• Let be a torus of rotation , ( ). Then there is an upwards and downwards unbounded transnormal function. ${\displaystyle T^{2}=\left\{(x,y,z)=((R+r\cos \theta )\cos \phi ,(R+r\cos \theta )\sin \phi ,r\sin \theta )\right\}\subset \mathbb {R} ^{3}}$ ${\displaystyle 0<r<R}$ ${\displaystyle f(x,y,z)=z}$

literature

• Qi Ming Wang: Isoparametric functions on Riemannian manifolds. I. Math. Ann. 1987, 277, no. 4, 639-646.
• Reiko Miyaoka: Transnormal functions on a Riemannian manifold. Differential Geom. Appl. 31 (2013), no. 1, 130-139.