Geometry

Let M and N be two oriented smooth manifolds, and F : M → N a local diffeomorphism. We say that F is oriented if for every positive top degree form on N, \(F^∗(\omega)\) is positive on M.

A subset \( X ⊂ \mathbb{R}^n\) is said to have measure zero if for every \(δ > 0\), there exists a countable cover of X by rectangles \((C_i)_{i≥0}\), such that \(\sum_{i\geq0}Vol(C_i) ≤ δ\).

Let \(A ⊂ \mathbb{R}^n\) be a closed rectangle and let \(f : A → \mathbb{R}\) be a bounded function. Assume that S = {x ∈ A|f not continuous at x} has measure zero. Then f is integrable.

 A subset \(D ⊂ \mathbb{R}^n\) is called a domain of integration if D is closed bounded and its boundary ∂D has measure zero.

Let D be a domain of integration and A a closed rectangle containing A. Let \( f : D → \mathbb{R}\) be a continuous function. Let \(\tilde{ f }: A → \mathbb{R}\) be the extension by zero. Then \(S( \tilde{f}) ⊂ ∂A\) has measure zero and we can set \(\int_Df := \int_A \tilde{ f}\). It is easy to see that this is independent of the choice of A. This is called the integral of f on D.

Let D and E be domains of integrations in Rn and let \(F : D → E\) be a smooth map inducing a diffeomorphism on the interior \(D^0 \xrightarrow{\sim} E^0\). Let \(g : E →\mathbb{R}\) be a continuous function. Then \(\int_E g = \int_D |det(DF)|\cdot g\circ F\) where DF is the Jacobian matrix.

A topoolgical manifold with boundary is a topological space which is locally homeomorphic to \(\mathbb{R}_+×\mathbb{R}^{n−1} = \mathbb{H}^n\). If M is a topological manifold with boundary, its boundary \(∂M\) is defined as the union of \(φ^{−1}({0}×\mathbb{R}^{n−1}∩φ(U))\) for all the charts \((U,φ)\) of M. A smooth manifold with boundary is a topological manifold with boundary endowed with a smooth atlas. If M is a smooth manifold with boundary, \(∂M\) is a smooth manifold.

Let M be a smooth oriented n-manifold with boundary. Let \(\omega\) be a compactly supported (n − 1)-form on M. Then \(\int_Md \omega= \int_{\partial M} \omega|_{∂M}\).

Let V be a finite dimensional vector space over \(\mathbb{R}\). An inner product on V is a symmetric bilinear form \(<.,. >: V ×V →\mathbb{R}\) which is definite-positive, i.e.m such that \(< v,v >> 0\), for every nonzero v ∈ V .

Let (V,<,>) be an inner product space. Given v ∈ V, we set \(∥v∥ = \sqrt{\langle v,v\rangle}\), which is called the length of v. Given two nonzero vectors \(v,w ∈ V\), the angle between v,w is defined as the unique \(θ ∈ [0,π]\) such that \(cosθ = \frac{\langle v,w \rangle}{∥v∥ · ∥w∥}\) . The right hand side belongs to [−1,1].

Let M be a smooth manifold and p : E → M a vector bundle on M. A symmetric bilinear form on E is a smooth section of B of \((E^V)^{⊗2} → M\), the bundle of bilinear forms on E, such that, for every p ∈ M, the bilinear fomr Bp : Ep × Ep → R is symmetric. 

Let M be a smooth manifold. A Riemann metric on M is an inner product ⟨., .⟩ on the tangent bundle TM of M. A Riemannian manifold is a pair consisting of a smooth manifold and a Riemannian metric on it.

Let (M,g) be an oriented Riemannian n-manifold. There is a unique n-form, denoted by \(dV_g\), on M, called the Riemannian volume form, characterized by one of the following two properties:

1. \(dV_g \) is positive with respect to the orientation on M, and has length equal to 1.

2. If \((x_1,..,x_n)\) are oriented loval coordinates, then \(dV_g = \sqrt{det(g_{ij})}dx_1 ∧ ... ∧ dx_n\)

Let (M,g) be an oriented Riemannian manifold (possibly with boundary). Assume that M is compact. We define the volume of (M,g) to be: \( Vol_g(M) = \int_M dV_g\)

Let F : N → M be smooth map between smooth manifolds. We say that F is an immersion if for every p ∈ M the induced map \(dF_p : T_pN → T_{F(p)}M\) is injective.

Let (M,g) and (N,h) be two Riemannian manifolds. A (local) isometry F : M →N is a (local) diffeomorphism sucht that \(F^∗(h) = g\).

Let M be a smooth manifold and let \(E \xrightarrow{p} M\) be a vector bundle. A connection ∇ on E is a the map \(∇: C^\infty(M;TM)×C^\infty(M;E)→C^\infty(M;E)\),  \((X,m) \mapsto ∇_Xm\) satisfying the following assumptions:

(i) For m fixed, \(m \mapsto ∇_Xm\) is \(C^\infty\)-linear, i.e., \(∇_{f_X+g_Y}m = f∇_Xm+g∇_Ym\).

(ii) For X fixed, \(m \mapsto ∇_Xm\) is R-linear and satisfies the following form of the Leibniz rule: \(∇_X(fm) = f∇_Xm+X(f)m\).

Let E → M be a vector bundle. Assume that E is endowed with a connection ∇ and an inner product g. We say that ∇ is compatible with g if for all \(X ∈ C^\infty(M,TX)\), \(v, w ∈ C^\infty(M,E)\), \(∇_Xg(v,w) = g(∇_Xv,w)+g(v,∇_Xw)\) which is again a form of the Leibniz rule.

 The curvature of ∇ is the map \(∧^2TM → End(M)\) of vector bundle send \(X_1∧X_2\) to \(∇_{X_1} ◦∇_{X_2} −∇_{X_2} ◦∇_{X_1} −∇_{[X_1,X_2]}\). This assignment is indeed linear in \(C^\infty(M)\) and well-defined.

Let M be a smooth Riemannian manifold, with Riemannian metric g. There is a unique connection ∇ on TM, calle the levi-civita connection, satisfying the following conditions:

(i) ∇ is compatible with the inner product g.

(ii) For X,Y vector fields, \(∇_XY −∇_YX = [X,Y]\)

Let M be a topological manifold.  A coordinate chart (or simply chart) on M is a pair \((U,φ)\), where U is an open subset of M and \(φ : U → φ(U)\) is a homeomorphism from \(U\) to an open subset \(φ(U) ⊂ \mathbb{R}^n\)

A topological space X is said to be locally connected (resp. locally path-connected) if every point of X has a cofinal system of open neighborhood which are connected (resp locally path-connected).

A subset A of a topological space X is said to be precompact if its closure \(A⊂X\) is compact.

 Let X be a topological space. We say that X is paracompact if every open covering of X admits a refinement which is locally finite.

Let X be a topological space. A family \((A_i)_{i∈I }\)of subsets in X is said to be locally finite if every point in X admits to an open neighborhood which meets only finitely many of the \(A_i\)

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.