The Unapologetic Mathematician

Mathematics for the interested outsider

Regular and Critical Points

Let f:M^m\to N^n be a smooth map between manifolds. We say that a point p\in M is a “regular point” if the derivative f_{*p} has rank n; otherwise, we say that p is a “critical point”. A point q\in N is called a “regular value” if its preimage f^{-1}(q) contains no critical points.

The first thing to notice is that this is only nontrivial if m\geq n. If m<n then f_{*p} can have rank at most m, and thus every point is critical. Another observation is that is q\notin f(M) then q is automatically regular; if its preimage is empty then it cannot contain any critical points.

Regular values are useful because of the generalization of the first part of the implicit function theorem: if q is a regular value of f:M\to N, then A=f^{-1}(q)\subseteq M is a topological manifold of dimension m-n. Or, to put it another way, A is a submanifold of “codimension” n=\dim(N). Further, there is a unique differentiable structure for which A is a smooth submanifold of M.

Indeed, let (V,y) be a coordinate patch around q with y(q)=0. Given p\in A, pick a coordinate patch (U,x) of M with x(p)=0. Let \pi_1:\mathbb{R}^m\to\mathbb{R}^n be the projection onto the first n components; let \pi_2:\mathbb{R}^m\to\mathbb{R}^{m-n} be the projection onto the last m-n components; an let \iota_2:\mathbb{R}^{m-n}\to\mathbb{R}^m be the inclusion of the subspace whose first m components are 0.

Now, we can write down the composition y\circ f\circ x^{-1}. Since this has (by assumption) maximal rank at 0\in\mathbb{R}^m, the implicit function theorem tells us that there is a coordinate patch (W,h) in a neighborhood of 0 such that y\circ f\circ x^{-1}\circ h=\pi_1\vert_W. So we can set \tilde{W}=\pi_2(W), which is open in \mathbb{R}^{m-n}, and get

\displaystyle y\circ f\circ x^{-1}\circ h\circ\iota_2\vert_{\tilde{W}}=\pi_1\circ\iota_2\vert_{\tilde{W}}=0

Setting z=x^{-1}\circ h\circ\iota_2\vert_{\tilde{W}} we conclude that z(\tilde{W})\subseteq A, since all these points are sent by f to the preimage y^{-1}(0)=q.

Now we claim that z(\tilde{W}) is not just any subset of A, but in fact z(\tilde{W})=A\cap x^{-1}(h(W)). Clearly z(\tilde{W}) is contained in this intersection, since

\displaystyle z(\tilde{W})=x^{-1}(h(\iota_2(\tilde{W})))=x^{-1}(h(W\cap(0\times\mathbb{R}^{m-n})))

On the other hand, if \tilde{p} is in this intersection, then \tilde{p}=x^{-1}(h(u)) for a unique u\in W — unique because x and h are both coordinate maps and thus invertible — and we have

\displaystyle0=y(f(\tilde{p}))=y(f(x^{-1}(h(u))))=\pi_1(u)

meaning that the first n components of u must be 0, and thus u\in(0\times\tilde{W}). Thus \tilde{p}\in z(\tilde{W}).

Therefore z maps \tilde{W}\subseteq\mathbb{R}^{m-n} homeomorphically onto a neighborhood of p\in A in the subspace topology induced by M. But this means that (z(\tilde{W}),z^{-1}) acts as a coordinate patch on A! Since every point p\in A can be found in some local coordinate patch, A is a topological manifold. For its differentiable structure we’ll just take the one induced by these patches.

Finally, we have to check that the inclusion \iota:A\to M is smooth, so A is a smooth submanifold — that its differentiable structure is compatible with that of M. But this is easy, since at any point p we can go through the above process and get all these functions. We check smoothness by using local coordinates x on M and z^{-1} on A, concluding that x\circ\iota\circ(z^{-1})^{-1}=h\circ\iota_2, which is clearly smooth.

April 21, 2011 Posted by | Differential Topology, Topology | 6 Comments

   

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