Ngô Quốc Anh

December 17, 2009

R-G: Defining function

Filed under: Riemannian geometry — Ngô Quốc Anh @ 15:58

Let M be a smooth manifold of dimension n.

Definition. If S \subset M is an embedded submanifold, a smooth map \Psi : M \to N such that S is a regular level set of \Psi is called a defining map for S. In other words,

\displaystyle S=\Psi^{-1}(c)

for some point c \in N. In particular, if N=\mathbb R^{n-p} (so that \Psi is a real-valued or vector-valued function), it is usually called a defining function.

Example 1. The sphere \mathbb S^n is an embeded submanifold of \mathbb R^{n+1}. The sphere is easily seen to be a regular level set of the function f:\mathbb R^{n+1} \to \mathbb R given by f(x)=|x|^2 since df=2 \sum_i x^i dx^i vanishes only at the origin.

Definition. More generally, if U is an open subset of M and \Psi:U \to N is a smooth map such that S \cap U is a regular level set of \Psi, then \Psi is called a local defining map (or local defining function) for S.

Example 2. The smooth map X:\mathbb R^2 \to \mathbb R^3 given by

\displaystyle X(\varphi, \theta)=\big( (2+\cos \varphi) \cos \theta, (2+\cos \varphi) \sin \theta, \sin \varphi\big)

is an immersion of \mathbb R^2 into \mathbb R^3 whose image, denoted by D, is the doughnut-shaped shape surface obtained by revolving the circle (y-2)^2+z^2=1 around the z-axis, a point (x,y,z) is in D if and only if it satisfies (r-2)^2+z^2=1 where r=\sqrt{x^2+y^2} is the distance from the z-axis. Thus D is the zero set of the function \Psi(x,y,z)=(r-2)^2+z^2-1 which is smooth on \mathbb R^3 minus the z-axis. A straightforward computation shows that $lated d\Psi$ does not vanish on D, so \Psi is a global defining function for D.

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