Randell’s Submanifolds (2.3) = Boothby’s Regular submanifold (III.5):

If K is a submanifold of M, then K has the subspace topology. If (ϕ_{α}, U_{α}) is a chart for M such that ϕ_{α}(U_{α}) = Π_{1}^{m}(−ϵ, ϵ) and ϕ_{α}(U_{α} ∩ K) = Π_{1}^{k}(−ϵ, ϵ) × Π_{k+1}^{m}{0} , then (ϕ_{α}|_{U_{α}∩K}, U_{α} ∩ K), ϕ_{α}|_{U_{α}∩K} : U_{α} ∩ K → Π_{1}^{k}(−ϵ, ϵ) is a chart for K.

Prop: If U^{open} ⊂ M^{m}, then U is an m-dimensional submanifold 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 the hypothesis K is a submanifold of M with K ⊂ M.

Prop: If f : N → M is smooth and if H is a submanifold of N, then f : H → M is smooth

Ex: Find a counterexample to the above if we replace the hypothesis H is a submanifold of N with H ⊂ N.

Prop: If f : N → M is smooth and if K is a submanifold of M and if f(N) ⊂ K, then f : N → K is smooth.

Ex: Find a counterexample to the above if we replace the hypothesis K is a submanifold of M with K ⊂ M.

2 Boothy III.6 = Randell Chapter 1.3 Defn: G is a topological group if

1.) (G, ∗) is a group 2.) G is a topological space. 3.) ∗ : G × G → G, ∗(g_{1}, g_{2}) = g_{1} ∗ g_{2}, and In : G → G, In(g) = g^{−1} are both continuous functions.

Defn: G is a Lie group if 1.) G is a group 2.) G is a smooth manifold. 3.) ∗ and In are smooth functions.

Ex: Gl(n, R) = set of all invertible n × n matrices is a Lie group:

1.) (Gl(n, R), matrix multiplication) is a group 2.) (Gl(n, R) is a smooth manifold. 3.) ∗(Gl(n, R) × (Gl(n, R) → (Gl(n, R),

∗(A, B) = AB and In : (Gl(n, R) → (Gl(n, R) In(A) = A^{−1} are smooth functions.

Ex: (C − {0}, ·), is a Lie group. Thm: If G is a Lie group and H is a submanifold, then H is a Lie group. Ex: (S^{1}, ·) Ex: G_{1}, G_{2} lie groups implies G_{1} × G_{2} is a lie group. Ex: T^{n} = S^{1} × ... × S^{1} is a Lie group. The following maps are diffeomorphisms: In : G → G, In(g) = g^{−1}. For a ∈ G, L_{a} : G → G, L_{a}(g) = ag R_{a} : G → G, R_{a}(g) = ga Ex: O(n) = {M ∈ GL(n, R) | M^{t}M = I} is a Lie group. Ex: Sl(n, R) = {M ∈ GL(n, R) | det(M) = 1} is a Lie group. Defn: F is a homomorphism of Lie groups if F is an algebraic homomorphism of Lie groups and F is smooth. Ex: F : GL(n, R) → R − {0}, F (M) = det(M) is a homomorphism.

Randell’s Submanifolds (2.3) = Boothby’s Regular submanifold (III.5):

K ⊂ N is a k-submanifold of N if ∀p ∈ K, there exists, Suppose f : N → M is smooth and has constant rank k. If q ∈ M, then f^{−1}(q) is a submanifold of N of dimension n − k.

Proof: Let p ∈ f^{−1}(q). By the rank theorem, Ex: F : (R, +) → (S^{1}, ·), F (t) = e^{2πit} is a homomorphism.

Ex: F : (R^{n}, +) → (T^{n}, ·), F (t_{1}, ..., t_{n}) = (e^{2πit_{1}}, ..., e^{2πit_{n}} is a homomorphism.

Thm: If F : G_{1} → G_{2} is a homomorphism of Lie groups, then

1.) rank(F) is constant.

2.) kernel of F = F^{−1}(e) is a closed submanifold

3.) F^{−1}(e) is a Lie group.

4.) dim(ker F) = dim(G_{1}) − rank(F)

Thm: If H is a submanifold and an algebraic subgroup of G, then H is closed in G.

Defn: G = group, X = set. G acts on X (on the left) if ∃σ : G × X → X such that

1.) σ(e, x) = x ∀x ∈ X
2.) σ(g_{1}, σ(g_{2}, x)) = σ(g_{1}g_{2}, x)

Notation: σ(g, x) = gx. Thus 1) ex = x; 2) g_{1}(g_{2}x) = (g_{1}g_{2})(x).

If G is a Lie group and X is a smooth manifold, then we require σ to be smooth.

Defn: The orbit of x ∈ X = G(x) = {y ∈ X | ∃g such that y = gx} Note: 1.) x ∈ G(x) 2.) If G(x) ∩ G(y) ≠ ∅, then G(x) = G(y) Thus we can use an action of G to partition X into disjoint subsets. Defn: If G acts on X, then X/G = X/ ∼ where x ∼ y iff y ∈ G(x) iff ∃g such that y = gx. If X is a topological space, then X/G = X/ ∼ is a topological 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 = R^{2}, σ((n, m), (x, y)) = (n + x, m + y). M/G = Ex: G = (Z_{2}, +), M = S^{n}, σ(0, x) = x, σ(1, x) = −x, . M/G =