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.

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.