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Linear Algebraic Underpinnings. a Generic Linear Map a : V → W Between finite-Dimensional Vector Spaces Has Maximal Rank

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Linear Algebraic Underpinnings. a Generic Linear Map a : V → W Between finite-Dimensional Vector Spaces Has Maximal Rank Linear algebraic underpinnings. A generic linear map A : V → W between finite-dimensional vector spaces has maximal rank. If dim(V ) ≤ dim(W ) then maximal rank is equivalent to A being injective. If dim(V ) ≥ dim(W ) then maximal rank is equivalent to A being surjective. Generic here means that the set of maximal rank maps is open and dense within the vector space hom(V, W ) of all linear maps. In particular , if we start with any A0 and perturb it in almost any direction, we will get a maximal rank map. Basic normal form theorems in linear algebra assert the following. (Injective case) If F : V → W is an injective linear map then there are LINEAR coordinates xi on V and linear coordinates ya on W such that relative to these coordinates F is given by yi = xi, i ≤ k, and yj = 0, j > k, i.e F (x1, . , xk) = (x1, . , xk, 0,..., 0) (∗ING) . (Surjective case) If A is an surjective linear map then there are LINEAR coordi- nates xi on V and linear coordinates ya on W such that relative to these coordinates F is given by yi = xi, i ≤ k i.e F (x1, . , xk) = (x1, . , xk, 0,..., 0) (∗SURJ) . These will also be normal forms in the nonlinear case. Immersions and submersions together form the class of maps between manifolds whose differentials are everywhere of maximal rank. The generic map between two manifolds will be either an immersion or submersion at most of its points. Here are the formal definitions. Definitions. Let F : M → N be a smooth map between manifolds. Recall that for p ∈ M the differential of F at p is written F∗p and is a linear map TpM → TF (p)N. Definition. F is called an immersion if its differential is everywhere injective, i.e. if ker(F∗p = 0 for all p ∈ M. F is called a submersion if its differential is everywhere onto : i.e. if im(F∗p) = TF (p)N for all p ∈ M. It makes sense to say a map is ‘an immersion at p’ or “a submersion at p”. Be- cause being maximal rank is an open condition on matrices, if a map is of maximal rank at p, then it is of maximal rank in some nbhd of p. Theorems for immersions. If F : M k → N n is an immersion, so that k ≥ n, and if p ∈ M is a point then there exist coordinates ya centered at F (p) ∈ N, and xi centered at p ∈ M, such that relative to these coordinates F is given by the normal form (*INJ): . Theorems for submersions. If F : M k → N n is an immersion, so that k ≥ n, and if p ∈ M is a point then there exist coordinates ya centered at F (p) ∈ N, and xi centered at p ∈ M, such that relative to these coordinates F is given by the normal form (*SURF). These theorems say that if F ’s differential is of maximal rank at a point p, then there exist local coordinates for the domain and range so that in these coordinates F looks precisely like its linearization does. These two theorems are special cases, and by far the most important cases, of what Lee calls the “ rank theorem” (theorem 7.8). They are all proved as applications of the Inverse Fn Thm. 1 2 F is called an embedding if it is • (a)an immersion, • (b)is one-to-one, • (c) and is a homeomorphism between M and F (M), the latter being en- dowed with the topology induced from N Topological theorem. If M is compact, we can dispense with (c) in the definition of embedding. In other words, for compact M, (a) and (b) imply (c). Sketch proof: A continous one-to-one map between compact Hausdorff spaces has a continuous inverse. Now, let’s forget about maps, and just talk about subsets. What does it mean for a SUBSET of Rn to an embedded submanifold? (Or a subset of N.) Definition: A subset Σ of RN is called an embedded submanifold if it can be given a manifold structure in such a way that the inclusion i :Σ → RN is a smooth embedding. REMINDER: The identity map between a space and itself need not be continuous if the space is given two different topologies say T, B. Eg: If T is the usual topology on R and B the discrete topology (every subset is an open subset) , then the identity map as a map (R, T) → (R, B) is not continuous. Pictures. Our definition is slightly different from Lee’s definition, but equivalent. See Lee’s Theorem 8. 3. Let us compare our definition, with Lee’s definition (p. 174) and the one you would find in an undergraduate differential geometry class (in the case k = 2, n = 3). Definition A. (“undergraduate”) A subset M k of Rn is a smooth manifold if every point p ∈ M is contained in an open ball U ⊂ Rn, such that there exists a smooth 1-1 immersion φ : Dk → Rn where Dk is the open k-disc, such that φ(D) = U ∩ M. Let us translate over to Lee’s definition, using his constant rank embedding theorem. By this theorem, and the fact that φ,is an immersion, so satisfies the hypothesis of the theorem, we know that there exist coordinates on ua on Dk, and xi on RN such that relative to these coordinates φ(u1, . uk) = (u1, . , uk, 0,..., 0). Lee calls such coordinates xi “slice coordinates”. Specifically, coordinates xi on N, defined in some nbhd U ⊂ N are called “slice coordinates” for the subset M ⊂ N, if M ∩ U = {xk+1 = 0, . , xN = 0}. We just proved above that what we call a “smooth submanifold of Rn admits slice coordinates. Theorem 0.1. A manifold in the sense of definition A is a manifold in our sense. Let M be a submanifold in the sense of definition A. By the above paragraph, for an point p we have slice coordinates xi for RN , centered at p. We need only check the overlap condition. So suppose that p lies in the intersection of the domains U and V of two sets of slice coordinates, say xi andx ˜i. The xi, i ≤ k and the x˜i, i ≤ k are both continuous coordinates for points of M in U ∩ V ∩ M. But are they smoothly related? Being coordinates on RN , we know that there is a smooth invertible map F : RN − → RN taking the xi to thex ˜i; F ∗x˜i = xi. Since M ∩ U ∩ V is given by xi = 0, xi = 0, i > k and also by and byx ˜i = 0, i > k, it follows that, for points of M ∩ U ∩ V the coordinates xi, x˜i, i ≤ k are related by we havex ˜i = F i(x1, . , xk, 0, 0,..., 0), i > k, while xi = Gi(˜x1,..., x˜k, 0,..., 0) where G = (G1,...,Gn) is the inverse to F in U ∩ V . This establishes a manifold structure for M. QED 3 Embedding cpt M in an RN , Step 1. We can cover M by a ‘regular open cover’ Wa so that each Wa is the n −1 domain of a chart ψa : Wa → B3(0) ⊂ R , and such that the sets Ua = ψa (B1(0)) also cover M. Choose a partition of unity subordinate to {Wa} with ρa = 1 on Ua. Step 2. Choose a diffeomorphism of B3(0) into the sphere of radius 1 about the n+1 origin in R . (For example: shrink B3 to have radius 1 by the map x 7→ 1/3x and then project up to the upper hemisphere xn+1 > 0. n+1 Step 3. The maps ga = g ◦ ρaφa : M → R are smooth and well-defined on all of M. We have ga(p) = 0 if and only if p /∈ Wa, and we have that ga is 1-1 on Wa, since it is one-to-one in the radial direction, as represented by ρa. Step 4. Piece all the ga together into one map g = (g1, g2, . , gM ) where M is the cardinality of the cover. According to step 3, the map g is a 1-1 immersion of M into RN with N = (n + 1)M. Apply the topological lemma to get that g is an embedding..
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