Geometry

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

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 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.

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.

 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 \(A ⊂ \mathbb{R}^n\) 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 \( X ⊂ \mathbb{R}^n\) 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 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 smooth manifold M is said to be orientable if \(∧^{top}T^∗ M\) is trivial. An oriented smooth manifold is a pair consisting of smooth manifold together with an orientation.

Let M be a smooth manifold of dimension n. An orientation of M is the choice of positive bases of \(∧^nT^∗M.\) More precisely, we consider the set of nowhere top degree forms \(\omega\) on M and we say that \( \omega ∼ \omega′ \) if there exists a positive smooth function f such that \(w′ = f\cdot\omega\) . An orientation is an equivalence class for this relation.

An oriented vector space is a finite dimensional vector space V together with an orientation. A basis of V is said to be positively oriented if it belongs to the chosen orientation.

On every finite dimenional vector space V over \( \mathbb{R}\) there are exactly two equivalence classes of bases for the relation of having the same orientation. An orientation of V is a choice of one of these classes.

Let V be a finite dimensional vector space over \(\mathbb{R}\). Let \((e_1,...e_n)\) and \((e′ _1, ..., e′ _n)\) be two bases of V. We say that these two bases have the same orientation if \(\frac{e′ _1∧...∧e′_n} {e_1∧...∧e_n} \in \mathbb{R} \backslash \{0\}\)  is positive. Equivalently, if we write \(e_i'=\sum_{j=1}^n a_{ij}e_j\) then \(det(a_{ij}) > 0\).

Let M be a smooth manifold, and \(\omega\) a closed differential n-form, with n ≥ 1. Then \(\omega\) is locally exact, i.e., can be written locally as \(dη\), for \(η\) an n−1-form.

 A subset \(U ⊂ \mathbb{R}^n\) is said to be star-shaped if there exists a point \(x_0 ∈ U\) such that for every \(y ∈ U\), the segment \([x_0,y]\) is contained in U.

Let M be a smooth manifold and assume that \(U,V ⊂ M\) are open subsets such that \(M = U ∪V\). Then we have a short exact sequence of de Rham complexes \(0 → Ω^•(M) → Ω^•(U)⊕Ω^•(V) → Ω^•(U ∩V) → 0\). It induces a long exact sequence in de Rham cohomology \(... → H^{n−1}_{ dR} (U ∩V)→H^n _{dR}(M) →H^n _{dR}(U)⊕H^n_{ dR}(V) → H^n _{dR}(U ∩V) → ...\)

Let M be a smooth manifold. Then the obvious projection \([0,1] × M \xrightarrow{p} M\) induces an isomorphism \(H^n _{dR}(M) \xrightarrow{p^*} H^n _{dR}([0,1] × M)\) for all \(n ∈ \mathbb{Z}\).

Let \(f^•,g^• : K^• → L^•\) be two morphisms of complexes. A homotopy from \(f^•\) to \(g^•\) is a homotopy operator \( \{h_n\}_{n∈\mathbb{Z}}\) such that \( f^n −g^n =d^{n−1}_ L ◦h_n+h_{n+1}◦d^n_ K\) for all \(n ∈ \mathbb{Z}\). 

 Let \(K^•\) and \(L^•\) be two complexes of vector spaces. A homotopy operator from \(K^•\) to \(L^•\) is a sequence of maps \( h_n : K^n → L^{n−1}\). The associated morphism of complexes is: \(\{d^{n−1}_ L ◦h_n+h_{n+1}◦d^n _K\}_{n∈Z}\).

Let \(K^•\) and \(L^•\) be two complexes of vector spaces. A morphism of complexes is a graded morphism \(f^• : K^• → L^• \) commuting with the differentials i.e. \(d^n _L◦f^n = f^{n+1}◦d^n _K\) for all \(n ∈ \mathbb{Z}\). 

The de Rham cohomology of M is the cohomology of the de Rham complex. It is denoted by: \(H^n _{dR}(M) := H^n(Ω^•(M)) = ker(d : Ω(M)^n → Ω(M)^{n+1})/Im(d : Ω(M)^{n−1} → Ω(M)^n)\)

Let M be a smooth manifold. The de Rham complex of M is the complex \((Ω^•(M),d)\) of smooth differential forms with the exterior derivative.

The n-th cohomology of a complex \(K^•\) is defined by \(H^n(K) := ker(d : K^n → K^{n+1})/Im(d : K^{n−1} → K^{n})\) For this to make sense we need that \(d^2 = 0\).

A complex (of vector spaces) is a graded vector space \(K^• = {K^n}_{n∈Z}\) to gether with a degree-1 linear map \(K^• \xrightarrow{d}K^{•+1}\), such that \(d^2 = 0\).

Let M be a smooth manifold. There are unique operators \(d : Ω^∗(M) → Ω^{∗+1}(M)\), called exterior differentiation, satisfying the following properties:

(i) d is linear over R.

(ii) If \(ω ∈ Ω^k(M)\) and \(η ∈ Ω^l(M)\), then \(d(ω ∧η) = dω ∧η +(−1)^kω ∧dη.\)

(iii) \(d ◦ d = 0\).

(iv) For \(f ∈ Ω^0(M) = C^\infty(M)\), df is the 1-form defined in the previous chapter.

 Let M be a manifold. A differential k-form on M is a smooth section of the vector bundle \(\wedge^k T^∗M\). If we don’t require smoothness, we speak of rough k-forms. The \(C^\infty(M)\)-module of differential k-forms is denoted by \( Ω^k(M) := C^\infty(M;\wedge^k T^∗M.)\)

Let M be a smooth manifold. Let \( f ∈ C^\infty(M)\) and \(γ : [a,b] → M\) a curve segment. Then \(\int_{\gamma}df = f(γ(b)) −f(γ(a))\).

 If \(γ : [a,b] → M\) and \(\tilde{ γ} : [c,d] → M\) are curve segments, we say that \(\tilde{ γ}\) is a reparametrization of \(\gamma\) if there exists an increasing diffeomorphism \(φ : [c,d] → [a,b] \)such that \(γ ◦φ = \tilde{ γ}\).