Randell’s Submanifolds (2.3) = Boothby’s Regular sub-manifold (III.5):If K is a submanifold of M, then K has the subspacetopology.If (φα, Uα) is a chart for M such that φα(Uα) = Πm1(−ǫ, ǫ)and φα(Uα∩ K) = Πk1(−ǫ, ǫ) × Πmk+1{0} , then(φα|Uα∩K, Uα∩ K), φα|Uα∩K: Uα∩ K → Πk1(−ǫ, ǫ) is achart for K.Prop: If Uopen⊂ Mm, then U is an m-dimensional sub-manifold of M.Prop: If K is a submanifold of M, then i : K → M,i(k) = k, the inclusion map is smooth.Ex: Find a counterexample to the above if we replace thehypothesis K is a submanifold of M with K ⊂ M.Prop: If f : N → M is smooth and if H is a submanifoldof N, then f : H → M is smoothEx: Find a counterexample to the above if we replace thehypothesis H is a submanifold of N with H ⊂ N.Prop: If f : N → M is smooth and if K is a submanifoldof M and if f(N) ⊂ K, t hen f : N → K is smooth.Ex: Find a counterexample to the above if we replace thehypothesis K is a submanifold of M with K ⊂ M.12Boothy III.6 = Randell Chapter 1.3Defn: G is a topological group if1.) (G, ∗) is a group2.) G is a topological space.3.) ∗ : G × G → G, ∗(g1, g2) = g1∗ g2, andIn : G → G, In(g) = g−1are both continuous f unct ions.Defn: G is a Lie group if1.) G is a group2.) G is a smooth manifold.3.) ∗ and In are smooth functions.Ex: Gl(n, R) = set of all invertible n × n matrices is a Liegroup:1.) (Gl(n, R), matrix multiplication) is a group2.) (Gl(n, R) is a smooth manifold.3.) ∗(Gl(n, R) × (Gl(n, R) → (Gl(n, R),∗(A, B) = AB andIn : (Gl(n, R) → (Gl(n, R)In(A ) = A−1are smooth funct ions.3Ex: (C − {0}, ·), is a Lie group.Thm: If G is a Lie group and H is a submanifold, then His a Lie group.Ex: (S1, ·)Ex: G1, G2lie groups implies G1× G2is a lie group.Ex: Tn= S1× ... × S1is a Lie group.The following maps are diffeomorphisms:In : G → G, In(g) = g−1.For a ∈ G,La: G → G, La(g) = agRa: G → G, Ra(g) = gaEx: O(n) = {M ∈ GL(n, R) | MtM = I} is a Lie group.Ex: Sl(n , R) = {M ∈ GL(n, R) | det(M) = 1} is a Liegroup.Defn: F is a homomorphism of Lie groups if F is an alge-braic homomorphism of Lie groups and F is smooth.Ex: F : GL(n, R) → R − {0}, F (M) = det(M) is ahomomorphism.4Randell’s Submanifolds (2.3) = Boothby’s Regular sub-manifold (III.5):K ⊂ N is a k-submanifold of N if ∀p ∈ K, there exists,5Suppose f : N → M is smooth and has constant rank k.If q ∈ M, then f−1(q) is a submanifold of N of dimensionn − k.Proof: Let p ∈ f−1(q). By t he rank theorem,6Ex: F : (R, +) → ( S1, ·), F (t) = e2πitis a homomorphism.Ex: F : (Rn, +) → (Tn, ·), F (t1, ..., tn) = (e2πit1, ..., e2πitnis a homomorphism.Thm: If F : G1→ G2is a homomorphism of L ie groups,then1.) rank(F) is constant.2.) kernel of F = F−1(e) is a closed submanifold3.) F−1(e) is a Lie group.4.) dim(ker F ) = dim(G1) − rank(F )Thm: If H is a submanifold and an algebraic subgroup ofG, then H is closed in G.Defn: G = group, X = set. G acts on X (on the left) if∃σ : G × X → X such that1.) σ( e, x) = x ∀x ∈ X2.) σ( g1, σ(g2, x)) = σ(g1g2, x)Notation: σ(g, x) = gx.Thus 1) ex = x; 2) g1(g2x) = (g1g2)(x).If G is a Lie group and X is a smooth manifold, then werequire σ to be smooth.Defn: The orbit of x ∈ X =G(x) = {y ∈ X | ∃g such that y = gx}7Note:1.) x ∈ G(x) 2.) If G(x) ∩ G(y) 6= ∅, then G(x) = G(y)Thus we can use an action of G to partition X into disjointsubsets.Defn: If G acts on X, then X/G = X/ ∼ where x ∼ y iffy ∈ G(x) iff ∃g such that y = gx.If X is a topological space, then X/G = X/ ∼ is a topo-logical space with the quotient topology.When is X/G = X/ ∼ a manifold?Ex: G = ( Z, +), M = R, σ(n, x) = n + x.M/G =Ex: G = ( Z × Z, +), M = R2,σ((n, m), (x, y)) = (n + x, m + y).M/G =Ex: G = ( Z2, +), M = Sn, σ(0, x) = x , σ(1, x) = −x, .M/G