Chapter 1 all sections 1.3 Defn: M is locally Euclidean of dimension n if for all p ∈ M, there exists an open set U_{p} such that p ∈ U_{p} and there exists a homeomorphism f_{p} : U_{p} → V_{p} where V_{p} ⊂ R^{n}. (U_{p}, f) is a coordinate nbhd of p. Let q ∈ U ⊂ M. f(q) = (f_{1}(q), f_{2}(q), ...., f_{n}(q)) ∈ R^{n} are the coordinates of q. f_{i} is the ith coordinate function Defn 3.1: An n-manifold, M, is a topological space with the following properties: 1.) M is locally Euclidean of dimension n. 2.) M is Hausdorff. 3.) M has a countable basis. 1.4 Let upper half-space, H^{n} = {(x_{1}, x_{2}, ..., x_{n}) ∈ R^{n} | x_{n} ≥ 0}, ∂H^{n} = {(x_{1}, x_{2}, ..., x_{n−1}, 0) ∈ R^{n}} ∼ R^{n−1} M is a manifold with boundary if it is Hausdorff, has a countable basis, and if for all p ∈ U, there exists an open set U_{p} such that p ∈ U_{p} and there exists a homeomorphism f : U_{p} → V_{p} one of the following holds: i.) V_{p} ⊂ H^{n} − ∂H^{n} (p is an interior point) or ii.) V_{p} ⊂ H^{n} and f(p) ∈ ∂H^{n} (p is a boundary point). ∂M = set of all boundary points of M is an (n-1)-dimensional manifold. 1.5 Ex: P^{n}(R) = RP^{n} = RP^{n} = (R^{n} − {0})/(x ∼ tx) = n-dimensional real projective space. T (M) = ∪_{p∈M}T_{p}(M)

Ch 2: all sections

2.1

Defn: Let f : R^{n} → R^{m}. Define g_{ij} : R^{1} → R^{1}, g_{ij}(t) = f_{i}(x_{1}, ..., x_{j−1}, t, x_{j+1}, ..., x_{n}). If g is differentiable at a, then the partial derivative of f_{i} is defined by

\frac{∂f_{i}}{∂x_{j}} (a) = lim_{h→0} \frac{f_{i}(a_{1},...,a_{j−1},a_{j}+h,a_{j+1},...,a_{n})−f_{i}(a_{1},...,a_{j−1},a_{j},a_{j+1},...,a_{n})}{h}

= lim_{h→0} \frac{f_{i}(a+he_{j})−f_{i}(a)}{h}

Defn: Suppose A ⊂ R^{n}, f : A → R^{m}.

f is said to be differentiable at a point a if there exists an open ball V such that a ∈ V ⊂ A and a linear function T such that

lim_{h→0} \frac{||f(a + h) − f(a) − T (h)||}{||h||} = 0

lim_{x→a} \frac{||f(x) − f(a) − T (x − a)||}{||x − a||} = 0

OR equivalently,

f is differentiable at a if and only if there is a matrix T and a function ϵ : R^{n} → R^{m} such that lim_{h→0}ϵ(h) = 0 and

f(a + h) − f(a) = T (h) + ||h||ϵ(h)

Or equivalently, there exists an m-tuple, R(x, a) = (r_{1}(x, a), r_{2}(x, a), ..., r_{m}(x, a) such that lim_{x→a}||R(x, a)|| = 0 and

f(x) = f(a) + T (x − a) + ||x − a||R(x, a)

Defn: Let V be a nonempty open subset of R^{n}, f : V → R^{m}, p ∈ N. i.) f is C^{p} on V is each partial derivative of order k ≤ p exists and is continuous on V.

ii.) f is C^{∞} (or smooth) on V if f is C^{p} on V for all p ∈ N (f is smooth).

iii.) f is C^{ω} (or analytic) on V if for all a ∈ V, near a, each component function can be written as a power series (e.g. if f : R → R^{m}, f(x) = f(a)+f′(a)(x−a)+ \frac{f′′(a)}{2!}(x−a)^{2}+... (its Taylor series)).

2.2 Defn: The Jacobian matrix of f at a is

[ \frac{∂f_{i}}{∂x_{j}} (a) ]_{m×n} = [ \begin{matrix}\frac{∂f_{1}}{∂x_{1}} (a) & & ... & & \frac{∂f_{1}}{∂x_{n}} (a) \\ . & . & & & . \\ . & & . & & . \\ . & & & . & . \\ \frac{∂f_{m}}{∂x_{1}} (a) & & ... & & \frac{∂f_{m}}{∂x_{n}} (a) \end{matrix} ]

2.3 T_{a}(R^{n}) = {(a, x) | x ∈ R^{n}}, ϕ(ax) = x − a, canonical basis = {ϕ^{−1}(e_{i}) | i = 1, ..., n} Let x(t) : R → R^{n}, a C^{1} curve such that x(0) = a T_{a}(R^{n}) = {[x(t)] | x ∈ C^{1}, x(0) = a} where x(t) ∼ y(t) if x′_{i}(t) = y′_{i}(t) for t ∈ (−ϵ, ϵ) Let f([x(t)]) = x′(0) = (x′_{1}(0), ..., x′_{n}(0)), then [x(t)] + [y(t)] = f^{−1}(x′(0) + y′(0)) and α[x(t)] = f^{−1}(αx′(0)) Let f : R^{n} → R and let v ∈ R^{n} such that ||v|| = 1 The directional derivative of f at a in the direction of v is D_{v}f(a) = lim_{h→0} \frac{f(a+hv)−f(a)}{h} = D[f(a + tv)]_{0} = Df_{a}v = Df_{a} · v = (\frac{∂f}{∂x_{1}} , ..., \frac{∂f}{∂x_{n}})|_{a} · v = ∇f · v 2.4 C^{∞}(a) = {[f] : X ⊂ R^{n} → R ∈ C^{∞} | a ∈ domf} where f ∼ g if ∃U^{open} s.t. a ∈ U and f(x) = g(x)∀x ∈ U. If X_{a} = Σ_{i=1}^{n}ξ_{i}E_{ia}, then X_{a}^{∗} : C^{∞}(a) → R, X_{a}^{∗}(f) = Σ_{i=1}^{n}ξ_{i} \frac{∂f}{∂x_{i}} |_{a} Note E_{ia}^{∗} = \frac{∂}{∂x_{i}} |_{a} The directional derivative of f at a in the direction of X_{a} = X_{a}^{∗}(f). Let D(a) = {D : C^{∞}(a) → R | D is linear and satisfies the Leibniz rule } D ∈ D(a) is called a derivation j : T_{a}(R^{n}) → D(a), j(X_{a}) = X_{a}^{∗} is an isomorphism.

2.5 Defn: A vector field is a function, V : U → ∪_{a∈U}T_{a}(R^{n}), such that V(a) ∈ T_{a}(R^{n}) Defn: A vector field is smooth if its components relative to the canonical basis {E_{ia} | i = 1, ..., n} are smooth. Defn: A field of frames is a set of vector fields {V_{1}, ..., V_{2}} such that {V_{1}(a), ..., V_{2}(a)} forms a basis for T_{a}(R^{n}) for all a. Note we can turn a vector field into a derivation by making the following definition: If V(a) = α_{i}(a)E_{ia}, then V : C^{∞} → C^{∞}, V(f)(a) = Σ_{i=1}^{n}α_{i}(a)\frac{∂f}{∂x_{i}} (a) is a derivation. 2.6 F is a C^{r}-diffeomorphism if

(1) F is a homeomorphism

(2) F, F^{−1} ∈ C^{r} F is a diffeomorphism if F is a C^{∞}-diffeomorphism. 2.7 Rank of A = dim(span{r_{1}, ..., r_{m}}) = dim(span{c_{1}, ..., c_{m}})

= maximum order of any nonvanishing minor determinant.

Rank of F : U ⊂ R^{n} → R^{m} at x = rank of DF (x) F has rank k if F has rank k at each x. Chapter 3: 1 - 3 3.1 = Randell 1.1 Note: This is one place where Boothby’s and Randell’s definitions differ: Boothby’s atlas = Randell’s pre-atlas, Boothby’s maximal atlas = Randell’s atlas. On exams, I will use pre-atlas and maximal atlas when it makes a difference. Defn: (φ, U) is a chart or coordinate neighborhood if φ : U → U′ is a homeomorphism, where U is open in M and U′ is open in R^{n}. (φ, U) is a coordinate nbhd of p.

Let q ∈ U ⊂ M. φ(q) = (φ_{1}(q), φ_{2}(q), ...., φ_{n}(q)) ∈ R^{n} are the coordinates of q. φ_{i} is the ith coordinate function. Defn: Two charts, (φ, U) and (ψ, V) are C^{∞} compatible if the function φψ^{−1} : ψ(U ∩V) → φ(U ∩ V) is a diffeomorphism. Defn: A (pre) atlas or differentiable or smooth structure on M is a collection of charts on M satisfying the following two conditions:

i.) the domains of the charts form an open cover of M ii.) Each pair of charts in the atlas is compatible.

Defn: An atlas is a (maximal or complete) atlas if it is maximal with respect to properties i) and ii). A differential (or smooth or C^{∞}) n-manifold M is a topological n-manifold together with a maximal atlas. 3.2 Let ∼ be an equivalence relation on X. Let π : X → X/ ∼, π(x) = [x] = {y | y ∼ x} [A] = ∪_{a∈A}[a] Defn: ∼ is open if U^{open} ⊂ X implies [U] open in X/ ∼. 3.3 = Randell 1.2 Defn: Suppose f : W → N where W ⊂ M and N are smooth manifolds. f is smooth if for all p ∈ W, ∃ charts (ϕ, U) and (φ, V) and such that p ∈ U, f(p) ∈ V, f(U) ⊂ V and φ ∘ f ∘ ϕ^{−1} is smooth. Defn: f : M → N is a diffeomorphism if f is a homeomorphism and if f and f^{−1} are smooth. M and N are diffeomorphic if there exists a diffeomorphism f : M → N. 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 topological group 2.) G is a smooth manifold. 3.) ∗ and In are smooth functions.

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 topological group and X is a topological space, then we require σ to be continuous. 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} Defn: If G acts on X, then X/G = X/ ∼ where x ∼ y iff y ∈ G(x) iff ∃g such that y = gx. Defn: The action of G on X is free if gx = x implies g = e. Defn: G is a discrete group if

0.) G is a group.

1.) G is countable

2.) G has the discrete topology Defn: The action of G on M is properly discontinuous if ∀x ∈ M, ∃U^{open} such that x ∈ U and U ∩ gU = ∅ ∀g ∈ G. Randell 2.1 Let p ∈ M. A germ is an equivalence class [g] where g^{smooth} : U → N, for some U^{open} such that p ∈ U ⊂ M and if g_{i} : U_{i} → N, where p ∈ U_{i}^{open} ⊂ M, then g_{1} ∼ g_{2} if ∃V^{open} such that p ∈ V ⊂ U_{1} ∩ U_{2} and g_{1}(x) = g_{2}(x) ∀x ∈ V.

G(p, N) = {[g] | g^{smooth} : U → N, for some U^{open} such that p ∈ U ⊂ M} G(p) = G(p, R) Let α : I → M where I = an interval ⊂ R, α(0) = p. Note [α] ∈ G[0, M] Directional derivative of [g] in direction [α] =

D_{α}g = \frac{d(g ∘ α)}{dt}|_{t=0} ∈ R

T_{p}(M) = {v : G(p) → R | v is linear and satisfies the Leibniz rule } v ∈ T_{p}(M) is called a derivation Given a chart (U, ϕ) at p where ϕ(p) = 0, the standard basis for T_{p}(M) = {v_{1}, ..., v_{m}}, where v_{i} = D_{α_{i}} and for some ϵ > 0, α_{i} : (−ϵ, ϵ) → M, α_{i}(t) = ϕ^{−1}(0, ..., t, ..., 0) Suppose f^{smooth} : M → N, f(p) = q. The tangent (or differential) map, df_{p} : T_{p}M → T_{q}N, (df_{p}(v)) = v([g ∘ f]) for v ∈ T_{p}M, [g] ∈ G(q) I.e., df_{p} takes the derivation (v : G(p) → R) ∈ T_{p}M to the derivation in T_{q}N which takes the germ [g] ∈ G(q) to the real number v([g ∘ f]).