Geometry

A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained in some set of C.

Let M be a topological n-manifold. An atlas for M is a collection of charts \(\{(U_α,φ_α)\}_{α∈I}\) whose domains cover M, i.e., \((U_α)_{α∈I}\) is an open covering of M.

Let M be a topological n-manifold. A smooth atlas on M is an atlas \(\{(U_α,φ_α)\}_{α∈I}\)such that the following condition is satisfied: For all pairs \(α,β ∈ I\), the composition \(φ_α(U_α ∩U_β) \xrightarrow{φ^{−1}_α}(U_α ∩U_β) \xrightarrow{\varphi_\beta} φ_β(U_α ∩U_β)\) is smooth, i.e., given by an n-tuple of smooth functions on the open subset \(φ_α(U_α∩U_β) ⊂ \mathbb{R}^n\)

Let M be a topological manifold endowed with a smooth atlas \(\{(U_α,φ_α)\}_{α∈I}\) Let \(f : M → \mathbb{R}\) be a continuous function. We say f is smooth if for every \(α ∈ I, f|_{U_α} ◦ φ^{−1}_α :φ_α(U_α) → \mathbb{R}\) is a smooth function. More generally, if \(f : V → \mathbb{R}\) a function defined on an open subset \(V \subset M\), we say that f is smooth if for every \(α ∈ I, f|_{U_α∩V} ◦φ^{−1}_α :φ_α(U_α ∩V) →\mathbb{R}\) is a smooth function.

Let M be a topological n-manifold. A smooth atlas A is said to be maximal if it is not strictly contained in a smooth atlas. (Here, we view A as a set of charts).

 A smooth n-manifold is a pair (M,A) consisting of a topological n manifold M and a maximal smooth atlas A. A function f on an open of M is considered smooth (or C∞) if it is so with respect to the smooth atlas A.

 Let V,W be finite-dimensional real vector spaces, which we assume to be endowed with norms. Let \(U ⊂ V\) be open and let \(F : U → W\) be a map. We say that F is differentiable at a point \(p ∈ U\) if there exists a linear map \(L : V → W\) such that lim \(\lim_{v→0} \frac{∥F(p +v)−F(p)−L(v)∥_W}{ ∥v∥_V} =0\)

Supppose V,W,X are finite dimensional vector spaces. Let U ⊂ V and U′ ⊂W be open subsets. Let F : U → U′ and G : U′ → X be maps. If F is differentiable at p ∈ U and G is differentiable at F(p) ∈ U′, then G ◦ F is differentiable at p ∈U and we have \(D(G◦F)(p) = DG(F(p))◦DF(p)\).

 Let \(U ⊂ \mathbb{R}^n\) open and let \(f : U → \mathbb{R}\) be a real-valued function. For \(p =(p_1,...,p_n) ∈ U\) and any 1 ≤ j ≤ n, the j-th partial deriavtive of f at p is defined to be the ordinary derivative of f with respect to the variable \(x_j\) at \(p_j\).\(\frac{∂f}{ ∂x_j} (p) := \lim_{ h→0} \frac{f(p+he_j)-f(p)}{h}\) where \(e_j\) is the j-th vector in the standard basis of \(\mathbb{R}^n\).

 Let \(U ⊂ \mathbb{R}^n\) open and F : U → W a function valued in a finite dimensional vector space W. We say that F is continuously differentiable (or that F is of class \(C^1\)) if the functions \(\frac{∂F}{ ∂x_j}\) , for 1 ≤ j ≤ n, are defined on U and are continuous.

Let k ≥ 1. Let F : U → W as in the previous definition. We say that F is of class \(C^0\) if F is continuous. We say that F is of class \(C^k\) if the functions \(\frac{∂F}{ ∂x_j }\), for 1 ≤ j ≤ n are defined and are of class \(C^{k-1}\). We say that F is smooth ( or \(C^\infty\)) if F is of class \(C^k\) for all k's.

If F is \(C^2\) then \(\frac{∂^2F}{ ∂x_j∂x_k }= \frac{∂^2F}{∂x_k ∂x_j}\).

 Let \(U ⊂ \mathbb{R}^m\) and \(V \subset\mathbb{R}^n\) be two open subsets. A map F : U → V is said to be \(C^k\)-diffeomorphism if F is bijective, and \(F\) and \(F^{−1}\) are of class \(C^k\). When \( k = \infty\), we also say smooth diffeomorphism.

Let U,V be open subsets of \(\mathbb{R}^n\). Let F : U →V be a map of class \(C^1\). If DF(p) at some point p ∈ U, then there exists open neighborhoods \(U_0 ⊂ U\) of p and \(V_0 \subset V\) of F(p), such that \(F|_{U_0} : U_0 → V_0\) is a diffeomorphism.

Let \(U ⊂ \mathbb{R}^n×\mathbb{R}^m\) be an open subset and denote the standard coordinates on U by \((x_1,...,x_n,y_1,...,y_m)\). Let \(ϕ : U → \mathbb{R}^m\) be a smooth map, (a,b) ∈ U and \(c = ϕ(a,b)\). If the m×m matrix \((\frac{∂ϕ_i }{∂y_j} (a, b))_{ 1≤i,j≤m}\) is invertible, there exists open neighborhoods\( V_0 ⊂ \mathbb{R}^n, W_0 ⊂ \mathbb{R}^m\) of a and b, and a smooth function \(F : V_0 → W_0\) such taht \(ϕ^{−1}(c) ∩ (V_0 × W_0) = graph(F)\).

In other words, \(ϕ(x,y) = c \text{ for }(x,y) ∈ V_0 ×W_0\) if and only if y = F(x).

Let M,N be smooth manifolds. A map \(F : M → N\) is said to be smooth if for every point p ∈ M with image q = F(p), we can find smooth charts \((U,φ)\) and \((V,ψ)\) of M and N such that p ∈ U and F(U) ⊂ V, and such that the composition of \(φ(U) \xrightarrow{φ^{−1}} U \xrightarrow{F|_U} V \xrightarrow{ψ} ψ(V)\) is a smooth map in the usual sense.

Let M,N be smooth manifolds and let \((U_α)_{α∈I}\) be an open covering of M. Suppose that for each α we are given a smooth map\( F_α : U_α →N\) such that for α,β ∈ I, \(F_α|_{U_α∩U_β} = F_β|_{U_α∩U_β}\). Then there exists a unique map \(F : M →N\) such that \(F|_{U_α} =F_α\) for each α ∈ I.

Let M,N be smooth manifolds. A map \(F : M → N\) is said to be a diffeomorphism if it is a bijection and if both \(F\) and \(F^{−1}\) are smooth.

Given real numbers \(r_1,r_2\) with \(r_1 < r_2\), there exists a smooth function \(h : \mathbb{R} → \mathbb{R}\) such that \(h(t) = 1\) for\( t ≤ r_1, 0 < h(t) < 1\) for
 \(r_1< t < r_2\), and \(h(t) = 0\) for \(t ≥ r_2\).
 

Let \(f: X → \mathbb{R}\) be a function on a topological space X. We define the support of \(f\) as the closure of the set of points where \(f\) is nonzero: \(supp(f) = \{p ∈ X|f(p)= 0\}\).

Let M be a topological space and let \((U_α)_{α∈I }\)be an open covering of M. A partition of unity subordinate to \((U_α)_{α∈I}\) is a family \((ψ_α)_{α∈I}\) of real-valued continuous functions on M with the following properties:

1.) \(0 ≤ ψ_α(x) ≤ 1\) for al α ∈ I and all x ∈ M.

2.) \(supp(ψ_α) ⊂ U_α \)for each α ∈ I.

3.) The family of supports \(supp(ψ_α)_{α∈I}\) is locally finite, i.e. every point of M has an open neighborhood intersecting \(supp(ψ_α)\) only for finitely many α’s.

4.) For every x ∈ M, \(\sum_{\alpha in I}ψ_α(x) = 1\)

Let M be a smooth manifold and \((U_α)_{α∈I}\) an open covering of M. Then there exists a smooth partition of unity subordinate to \((U_α)_{α∈I}\).

Let M be a topological space, \(A ⊂ M\) a closed subset, \(U ⊂ M\) an open subset containing A. A continuous function \(ψ : M → \mathbb{R}\) is said to be a bump function for A supported in U if \(0 ≤ ψ(x) ≤ 1\) for all \(x ∈ M, ψ(x) = 1\) for all \(x ∈ A\), and \(supp(ψ) ⊆ U\).

Let M be a smooth manifold. For any closed subset A ⊂ M and any open subset U ⊂ M containing A, there is a smooth bump function for A supported in U.

Let M,N be smooth manifolds, and let A ⊂ M be any subset. We say that a map \(F : A → N\) is smooth on A, if it has locally smooth extensions, i.e., for every \(p ∈ A\) there is an open neighborhood \( U ⊂ M\) of x, a smooth map  \(\tilde{F}_U : U → N\), such that \(F|_{A∩U} = \tilde{F}_U|_{A∩U}\).

Let M be a smooth manifold, \( A ⊂ M\) a closed subset, and \(f : A → \mathbb{R}\) a smooth function. For any subset U containing A, there is a smooth function \(\tilde{f }: M → \mathbb{R}\) supported in U and such that \(\tilde{ f}|_A = f.\)

Let M be a topological manifold (or space, but has to be Hausdorff). An exhaustion function for M is a continuous function \(f : M → \mathbb{R}\) such that \(f^{−1}((−∞,c])\) is a compact subspace for all \(c ∈ \mathbb{R}\). We call this subset the level subset.

Every smooth manifold admits a smooth positive exhaustion function.

\(D_v(f):=\frac{d}{dt}|_{t=0}f(p+tv)=\lim_{t \rightarrow 0}\frac{f(p+tv)-f(p)}{t}\). Note that \(D_v:C^{\infty}(U)\rightarrow \mathbb{R}\)

Let \(U ⊂ \mathbb{R}^n\) be open and \(p ∈ U\) a point. A derivation at p is a linear map \(∂ : C^\infty(U) →\mathbb{R}\) satisfying the following form of the Leibniz rule:\( ∂(fg) = ∂(f)·g(p) +f(p)\cdot\partial g(p)\). We denote by \(Der_p(U)\) the set of all derivations at p.

Let \(U \subset \mathbb{R}^n \) open, \(p\in U\) and \(\partial:C^\infty(U)\rightarrow\mathbb{R}\) a derivation at p.

1.) If f is constant then \(\partial f=0\)

2.) If \(f(p)=g(p)=0\) then \(\partial (fg)=0\)

Let \(U ⊂ \mathbb{R}^n\) be an open subset, and let \(p ∈ U\) be a point. Let f be a smooth function on U. If W is a convex subset of U containing p, then for all x ∈ W we have: \( f(x) = \sum_{|I|\leq k}\frac{D^If(p)}{I!}(x-p)^I+R_k\)where \(R_k\) is a smooth function given by \(R_k = \sum_{|I|≥k+1} \frac{k +1}{ I! }(x−p)^I \int_0^1 (1 −t)^k D^If(p+t(x−p)) dt\).

Let M be a smooth manifold, (U,φ) a smooth chart, p∈U. Then the tangent vectors \(\frac{∂}{ ∂x_1} |_p ,...,\frac{∂}{ ∂x_n} |_p\) are called the coordinate vectors at p associated with the given chart (U,φ). They form a basis of \(T_pM\).

For each p ∈ M, there is an associated linear map, the differential of F at p \(dF_p : T_pM →T_{F(p)}N\) defined as follows; Given a derivation \(v : C^\infty(M) →\mathbb{ R}\) at p, its image by \(dF_p\) is the derivation \(ω : C^\infty(N) →N\) at F(p) given by \(ω(g) := v(g◦F)\).