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Home , Local diffeomorphism, Submanifold, Submersion (mathematics)

, Immersions and Submersions (Ravi Banavar, Systems and Control, IIT Bombay.)

When is a subset S of an n-dimensional smooth M called a of M ? We answer this question and address a few related issues here. Definition 0.1 Submanifold

If for each element p ∈ S, there exists a chart (Up, φ) (note: Up is an in M and φ T k (n−k) is a homeomorphism) such that φ maps Up S to a subset of R × 0 , then S is said to be a submanifold of M of dimension k. Example 1: Look at the unit circle S1 as a subset of R2. Let the open sets (or the topology) on R2 be the open discs, and the mapping φ be the identity map to R2. So

2 2 φ(·): R → R , φ(x, y) = (x, y) The points on the unit circle centred at 0 satisfy

1 2 2 p = (xp, yp) ∈ S ⇒ xp + yp = 1 and get mapped to R2 through the homeomorphism φ as q 2 φ(xp, yp) = (xp, yp) = (xp, ± (1 − xp)) The ± sign in the right-most term indicates either a + or − value depending on the quadrant.

Note that the coordinate mapping is a of just one variable xp. We shall now introduce a local diffeomorphism to ”zero” one of the coordinates. Define " # 2ψ 2η β(ψ, η) = (ψ2 + η2 − 1, η)[Dβ] = 0 1

Notice that β is a diffeomorphism in a region around (ψ0, η0) if and only if ψ0 6= 0. Around 1 1 a point p on S and an open set Up containing p such that (0, y) ∈/ (S ∩ Up) (to satisfy the above Jacobian condition), we write q q 0 4 2 2 φ (xp, yp) = β ◦ φ(xp, yp) = β(xp, ± (1 − xp)) = (0, ± (1 − xp))

Notice that the mapping β ◦ φ is to a one dimensional subspace (0, R1) of R2.

How does one decide if a level set on a manifold is a regular submanifold ?

There are a few results that could be employed. The theorem and the are two results that help. Before we do so, we define a few new concepts.

1 Immersions, submersions and critical values

Definition 0.2 (Submersion and ) If f : X → Y is smooth, then

1. If Tpf is onto for all p ∈ X, then f is called a submersion

2. If Tpf is one-to-one for all p ∈ X, then f is called a immersion

3. If f is an immersion and f is one-to-one, then f(X) is an immersed submanifold

Definition 0.3 (Regular point and critical point) Consider a smooth map f : X → Y

• A point p ∈ X is called a critical point of f if Tpf is not onto.

• A point p ∈ X is called a regular point of f if Tpf is onto.

• A point y ∈ Y is called a critical value if f −1(y) contains a critical point. Otherwise, y is called a regular value of f.

The Submersion Theorem If f : Xn → Y k is a smooth map, and y ∈ f(X) ⊂ Y is a regular value of f, then f −1(y) is a regular submanifold of X of dimension n − k.

• Example: f(x, y) = x2 + y2 − 1. Take f −1(0).

T • Show that O(n) = {A ∈ Mn(R)|A A = I} is a submanifold of Mn(R). What is its dimension ? Hint: Consider a map f : Rn×n → Sn×n (symmetric matrices). Let f(A) = AT A and examine the value I.

Consider the S2 in R3 and consider the homemorphism

Remark 1 For a smooth map f : N → M of , a level set f −1(c) is regular if and only if f is a submersion at every point of f −1(c). (on the lines of the submersion theorem.)

Example 1 Consider the smooth map f : R → R2, f(t) = (t2, t3). The map is one-to-one but the Jacobian " # 2t [Df](t) = 3t2 is not one-to-one at zero. Is the curve a submanifold of R2 ? No. The condition to be a submanifold fails at (0, 0). At all points p ∈ S (except 0), one can find an opens set Up containing p and a smooth submersion g such that Up ∩ S is a level set of g.

2 Example 2 Consider the smooth map f : R → R2, f(t) = (t2 − 1, t3 − t). The map is not one-to-one (f(−1) = f(1) = (0, 0)) but the Jacobian " # 2t [Df](t) = 3t2 − 1 is one-to-one everywhere.

Is the curve a submanifold of R2 ? No. The condition to be a submanifold once again fails at (0,0). At all points p ∈ S (except

0), one can find an opens set Up containing p and a smooth submersion g such that Up ∩ S is a level set of g.

Example 3 The figure-eight f : R → R2, f(t) = (cos t, sin 2t). The map is not one-to-one but the Jacobian Df(t) = (− sin t, 2 cos 2t) is one-to-one everywhere. However, note that if this map is restricted to the open interval (−π/2, 3π/2), then the map is one-to-one. It can also be verified that it is a submanifold of R2.

Example 4 (A spiral) Consider the smooth map f : R → R3, f(t) = (sin 2πt, cos 2πt, t). The map is one-to-one and the Jacobian

Df(t) = (2π cos 2πt, −2π sin 2πt, 1)

is one-to-one everywhere. At all points p ∈ S one can find an opens set Up containing p and 3 2 a smooth submersion g = (g1, g2): R → R such that Up ∩ S is a level set of g.

Definition 0.4 A smooth map f : N → M is called an embedding if

• it is a one-to-one immersion (recall the definition of an immersed submanifold.)

• the f(N) with the is homeomorphic to N under f.

Example 5 The figure-eight described by the map

f(t) = (cos t, sin 2t) − π/2 < t < 3π/2

is one-to-one and the Jacobian of this map

Df(t) = (− sin t, 2 cos 2t)

3 is one-to-one everywhere on the open interval (−π/2, 3π/2). Hence the figure-eight is the im- age of a one-to-one immersion. Hence it is an immersed submanifold of R2 with a topology and manifold structure induced from the open interval (−π/2, 3π/2) by f. Is the figure-eight a regular submanifold of R2 ? No, in the subspace topology of R2, the figure-eight is not even a manifold. Why ? (Hint: examine the homeomorphism condition around (0, 0).)

The tangent

Let us now prove that the tangent bundle TM of a manifold M is a manifold. We shall start with manifolds immersed in Rn.

Definition 0.5 A subset M of Rn is said to be a submanifold of dimension m (≤ n) of Rn n if for every p ∈ M there exists an open set Up ⊂ R containing p such that

−1 n−m n−m M∩Up = {f (c): where c ∈ R is a constant and f : Up → R is a smooth submersion}

We now show, from the perspective of the above definition, that the tangent bundle TM of M is a sub manifold. n n For every (p, vp) ∈ TM ⊂ R × R , there exists an open set Up × V , such that

˜−1 TM ∩ (Up × V ) = {f (˜c)},

where " # " # f c f˜ = c˜ = , Df 0 where Df denotes the differential of f (or the Jacobian). (Why ?)

Consider a smooth curve α(·): I → M, such that α(0) = p. Thenα ˙ (0) = vp is a tangent vector at p. Now, from f ◦ α(t) = c for a curve on M,

d (f ◦ α)(t) = 0 ⇒ Df(p) · α˙ (0) = 0 dt

Since f is a smooth submersion of (n − m) and it follows that [Df(p)]−1(0) = Rm.

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