• Prerequisites
  • Topology of Manifolds Intro
  • Algebras
  • Derivations
  • Equivalence Relations
  • On relation to Physics
• Manifolds
  • Atlases and Charts
  • Definition of manifold
  • Examples of manifolds
    • N-spheres
    • Products and n-torus
    • Real projective spaces
    • Complex projective spaces
    • Grassmannians
    • Complex Grassmannians
  • Oriented manifolds
  • Open subsets
  • Compact subsets
• Smooth Maps
  • Smooth functions on manifolds
  • Smooth maps between manifolds
  • Diffeomorphisms of manifolds
  • Examples of smooth maps
    • Products, diagonal maps
    • The diffeomorphism \(\mathbb R P^1 \cong S^1\)
    • The diffeomorphism \(\mathbb C P^1 \cong S^2\)
    • Maps to and from projective space
    • The Hopf fibration, a.k.a. the quotient map \(S^{2n+1} \rightarrow \mathbb C P^n\)
  • Submanifolds
  • Smooth maps of maximal rank
    • The rank of a smooth map
    • Local diffeomorphisms
    • Level sets, submersions
    • Example: The Steiner surface
    • Immersions
• The Tangent Bundle
  • Tangent map
    • Definition of the tangent map, basic properties
    • Coordinate description of the tangent map
  • Tangent spaces of submanifolds
    • Example: Steiner’s surface revisited
    • The tangent bundle
• Vector Fields
  • Vector fields as derivations
  • Vector fields as sections of the tangent bundle
  • Lie brackets
  • Related vector fields
  • Flows of vector fields
  • Geometric interpretation of the Lie bracket
  • Frobenius theorem
• Differential Forms
  • Review: Differential forms on \(\mathbb R^m\)
  • Dual spaces
  • Cotangent spaces
  • 1-forms
  • Pull-backs of function and 1-forms
  • Integration of 1-forms
  • 2-forms
  • k-forms
    • Definition
    • Wedge product
    • Exterior differential
  • Lie derivatives and contractions
    • Pull-backs
    • Integration of differential forms
    • Integration over oriented submanifolds
    • Stokes’ theorem
    • Volume forms
• De Rham Cohomology
• Overview of Riemannian Geometry
  • Covariant derivative and curvature
  • Curvature
• Lie Groups

Prerequisites

Topology of Manifolds Intro

The fundamental objects of study in differential geometry are manifolds. Roughly, an n-dimensional manifold is a mathematical object that “locally” looks like \(\mathbb R^n\). Manifolds in euclidean space are described with a regular level set, \(S = f^{-1}(a)\) which defines a smooth hypersurface \(S \subseteq R^n\). For example, the n-dimensional sphere described by: \[S^n = \{ (x^0, \dots, x^n) \in \mathcal R^{n+1} | (x^0)^2 + \dots + (x^n)^2 = 1\}.\] Another example is the 2-Torus, \(T^2\). Given real numbers \(r, R\) with \(0 < r < R\), take a circle of radius \(r\) in the \(x-z\) plane, with center at \((R,0)\), and rotate about the \(z\)-axis: \[T^2 = \{ (x,y,z) | (\sqrt{x^2 + y^2} - R)^2 + z^2 + r\}\]

The sphere, the torus, the double torus, triple torus, and so on are orientable surfaces, which essentially means that they have two sides which you might paint in two different colors. It turns out that these are all orientable surfaces, if we consider the surfaces intrinsically and only consider surfaces that are compact in the sense that they don’t go off to infinity and do not have a boundary (thus excluding a cylinder, for example).

Not all surfaces can be realized as ‘embedded’ in \(\mathbb R^3\); for non-orientable surfaces one needs to allow for self-intersections. This type of realization is referred to as an immersion: We don’t allow edges or corners, but we do allow that different parts of the surface pass through each other. An example is the Klein bottle, which is not possible to represent as a regular level set \(f^{-1}(0)\) of a function \(f\) since any suface has one side where \(f\) is positive and another side where \(f\) is negative.

The projective plane or projective space is denoted \(\mathbb{R} P^2\) and is defined as the set of all lines (i.e., 1-dimensional subspaces) in \(\mathbb{R}^3\). we can also think of \(\mathbb{R} P^2\) as the set of antipodal (i.e., opposite) points on \(S^2\). Splitting the points into those with distance \(< \epsilon\) from the equator and those \(\geq \epsilon\) produces a Mobius strip and a two-dimensional disc. Generating a smooth curve by gluing the boundary of a Mobius strip to the boundary of a disk is depicted in what’s known as Boy’s surface.

Another operation for surfaces, generalizing the procedure of ‘attaching handles’, is the connected sum Given two surfaces \(\sigma_1\) and \(\sigma_2\), remove small disks around given points \(p_1 \in \sigma_1\) and \(p_2 \in \sigma_2\), to create two surfaces with boundary circles. Then glue-in a cylinder connecting the two boundary circles, without creating edges. The resulting surface is denoted \(\sigma_1\#\sigma_2\).

It turns out that all closed, connected surfaces are obtained from either the 2-sphere \(S^2\), the Klein bottle, or \(\mathbb{R} P^2\), by attaching handles with the connected sum.

Algebras

An algebra (over the field \(\mathbb R\) of real numbers) is a vector space \(\mathscr{A}\), together with a multiplication (product) \(\mathscr{A} \times \mathscr{A} \rightarrow \mathscr{A}\), \((a,b) \mapsto ab\) such that

1. The multiplication is associative.

2. The multiplication map is linear in both arguments.

The algebra is called commutative if \(ab = ba\) for all \(a,b \in A\). A unital algebra is an algebra \(A\) with a distinguished element \(1_{\mathscr{A}} \in \mathscr{A}\) (called the unit), with \(1_{\mathscr{A}} a = a = a1_{\mathscr{A}}\) for all \(a \in \mathscr{A}\).

One can also consider non-associative product operations on vector spaces, most importantly one has the class of Lie algebras. Some examples include: The space of complex numbers which is a unital, commutative algebra. \(\mathbb H \cong \mathbb R^4\) of quaternions which is a unital, non-commutative algebra. The space of \(n\times n\) matricies which is a noncommutative unital algebra. Given a topological space \(X\), one has the algebra \(C(X)\) of continuous \(\mathbb R\)-valued functions.

A homomorphism of algebras \(\phi : \mathscr{A} \rightarrow \mathscr{A}'\) is a linear map preserving products: \(\phi(ab) = \phi(a)\phi(b)\). It is called an isomorphism of algebras if \(\phi\) is invertible. For the special case \(\mathscr{A}' = A\) , these are also called algebra automorphisms of \(\mathscr{A}\). Note that the algebra automorphisms form a group under composition.

Derivations

If \(dim A < \infty\), a derivation is an infinitesimal automorphism of an algebra.

Any given \(x \in A\) defines a derivation \(D(a) = [x,a] := xa-ax\). These are called inner derivations. If \(A\) is commutative (for example \(A = C ^\infty(M)\)) the inner derivations are all trivial. At the other extreme, for the matrix algebra \(A = \text{Mat}\mathbb R(n)\), one may show that every derivation is inner.

If \(A\) is a unital algebra, with unit \(1_A\), then \(D(1A) = 0\) for all derivations \(D\).

Given two derivations \(D_1,D_2\) of an algebra \(A\), their commutator \([D_1,D_2] = D_1 D_2 - D_2 D_1\) is again a derivation.

If the algebra \(A\) is commutative, then the space of derivations is a ‘left-module over \(A\)’. That is, if \(D\) is a derivation and \(x \in A\) then \(a \mapsto (xD)(a) := xD(a)\) is again a derivation

Equivalence Relations

[Omitted]

On relation to Physics

In Albert Einstein’s theory of General Relativity from 1916, space-time was regarded as a 4-dimensional ‘curved’ manifold with no distinguished coordinates. A local observer may want to introduce local \(xyz\) coordinates to perform measurements, but all physically meaningful quantities must admit formulations that are coordinate-free. At the same time, it would seem unnatural to try to embed the 4-dimensional curved space-time continuum into some higher-dimensional flat space, in the absence of any physical significance for the additional dimensions. Some years later, gauge theory once again emphasized coordinate-free formulations, and provided physics motivations for more elaborate constructions such as fiber bundles and connections. There are many subbranches of differential geometry, for example complex geometry, Riemannian geometry, or symplectic geometry, which further subdivide into sub-sub-branches.

Manifolds

Atlases and Charts

Manifolds will initially be described in intrinsic or manifestly coordinate-free terms. The basic feature of manifolds is the existence of ‘local coordinates’. The transition from one set of coordinates to another should be smooth.

The 2-sphere with the atlas given by stereographic projections onto the \(x-y\)-plane, and the 2-sphere with the atlas given by stereographic projections onto the \(y-z\)-plane, should be one and the same manifold \(S^2\). To resolve this, we will use the following notion of compatibility.

Note that \((U,\phi)\) is compatible with the atlas \(\mathscr{A} = \{(U_\alpha, \phi_\alpha) \}\) if and only if the union \(\mathscr{A} \cup {(U,\phi)}\) is again an atlas on \(M\). This suggests defining a bigger atlas, by using all charts that are compatible with the given atlas with the following lemma.

Definition of manifold

The charts \((U,\phi) \in A\) are called (coordinate) charts on the manifold \(M\). The countability condition is used for various arguments involving a proof by induction. The Hausdorff condition rules out some strange examples that don’t quite fit the idea of a space that is locally like \(\mathbb R^n\).

Examples of manifolds

N-spheres

The construction of an atlas for the 2-sphere \(S^2\), by stereographic projection, also works for the n-sphere \[S^n = \{ (x^0, \dots, x^n) \in \mathcal R^{n+1} | (x^0)^2 + \dots + (x^n)^2 = 1\}.\] Let \(U_{\pm}\) be the subsets obtained by removing \((\pm 1, 0,\dots, 0)\). Stereographic projection defines bijections \(\phi_{\pm} : U_{\pm} \rightarrow \mathbb R^{n}\), where \(\phi_{\pm} (x^0, x^1, \cdots, x^n) = (u^1, \cdots, u^n)\) with \(\displaystyle u^{i} = \frac{x^j}{1\pm x^0}.\)

Writing \(u = (u^1,\dots,u^n)\), the transition functions are given by, \[(\phi_- \circ \phi_+^{-1})(u) = \frac{u}{||u||^2}.\]

Products and n-torus

Given manifolds \(M,M'\) of dimensions \(m,m'\), with atlases \(\{(U_\alpha,\phi_\alpha) \}\) and \(\{(U_\beta,\phi_\beta)\}\), the cartesian product \(M \times M'\) is a manifold of dimension \(m + m'\). An atlas is given by the product charts \(U_\alpha \times U_\beta\) with the product maps \(\phi_\alpha \times \phi_\beta' : (x, x') \mapsto (\phi_\alpha(x), \phi_\beta'(x'))\). For example, the 2-torus \(T^2 = S^1 \times S^1\) becomes a manifold in this way, and likewise for the n-torus, \(T^n = S^1 \times \dots \times S^1\).

Real projective spaces

The n-dimensional projective space \(\mathbb R P^n\), is the set of all lines \(l \subseteq \mathbb R^{n+1}\). It may also be regarded as a quotient space, \[\mathbb R P^n = (\mathbb R^{n+1} \setminus \{0\}) / \sim\] \(\mathbb R P^n\) has a standard atlas, \(\mathscr{A} = {(U_0,\phi_0),\dots,(U)n,\phi_n)}\) defined as follows. For \(j = 0,\dots,n,\) let \(U_j = \{(x^0 : \dots : x^n) \in \mathbb R P^n | x^j \neq 0\}\) be the set for which the j-th coordinate is non-zero, and put \[\phi_j : U_j \rightarrow \mathbb R^n, (x^0 : \dots : x^n ) \mapsto \bigg ( \frac{x^0}{x^j} ,\dots, \frac{x^n}{x^j} \bigg).\]

Geometrically, viewing \(\mathbb RP^n\) as the set of lines in \(\mathbb R^{n+1}\), the subset \(U_j \subseteq \mathbb RP^n\) consists of those lines \(l\) which intersect the affine hyperplane \(H_j = \{x \in \mathbb R^{n+1} | x^j = 1\}\) and the map \(\phi_j\) takes such a line \(l\) to its unique point of intersection \(l \cap H_j\), followed by the identification \(H_j \cong \mathbb R^n\) (dropping the coordinate \(x^j = 1\)). In low dimensions, we have that \(\mathbb RP^0\) is just a point, while \(\mathbb RP^1\)is a circle.

Complex projective spaces

Similar to the real projective space, one can define a complex projective space \(\mathbb C P^n\) as the set of complex 1-dimensional subspaces of \(\mathbb C^{n+1}\). \[\mathbb C P^n = (\mathbb C^{n+1} \setminus \{0\})/ \sim\] Alternatively, letting \(S^{2n+1} \subseteq \mathbb C^{n+1} = \mathbb R^{2n+2}\) be the ‘unit sphere’ consisting of complex vectors of length \(||z|| = 1\), we have \[\mathbb C P^n = S^{2n+1} / \sim\] One defines charts \((U_j ,\phi_j)\) similar to those for the real projective space: \[\begin{aligned} U_j &= \{(z^0 : \dots : z^n) | z^j \neq 0\} \\ \phi_j : U_j \rightarrow \mathbb C^{2n},& \ \ (z^0 : \dots : z^n ) \mapsto \bigg ( \frac{z^0}{z^j} ,\dots, \frac{z^n}{z^j} \bigg).\end{aligned}\] The transition maps between charts are given by similar formulas as for \(\mathbb RP^n\) (just replace \(x\) with \(z\)). The transition maps are not only smooth but even holomorphic, making \(\mathbb C P^n\) an example of a complex manifold (of complex dimension \(n\)).

Grassmannians

The set \(Gr(k,n)\) of all \(k\)-dimensional subspaces of \(\mathbb R^n\) is called the Grassmannian of \(k\)-planes in \(\mathbb R^n\). As a special case, \(Gr(1,n) = \mathbb RP^{n-1}\). We will show that for general \(k\), the Grassmannian is a manifold of dimension \(dim(Gr(k,n)) = k(n-k)\). [Omitted]

Complex Grassmannians

Similar to the case of projective spaces, one can also consider the complex Grassmannian \(Gr \mathbb C(k,n)\) of complex \(k\)-dimensional subspaces of \(\mathbb C^n\) . It is a manifold of dimension \(2k(n-k)\), which can also be regarded as a complex manifold of complex dimension \(k(n-k)\).

Oriented manifolds

The notion of an orientation on a manifold will become crucial later, since integration of differential forms over manifolds is only defined if the manifold is oriented.

The spheres \(S^n\) are orientable. To see this, consider the atlas with the two charts given by stereographic projections. The Jacobian matrix \(D(\phi_-, \psi^{-1})_+)(u)\) has determinant of \(-||u||^{-2n}\) which is not an oriented atlas. To remedy this, simply compose one of the charts, with the map \((u_1,u_2,\dots,u_n) \mapsto (-u_1,u_2,\dots,u_n)\); then with the resulting new coordinate map \(\Tilde{\phi}_-\) the atlas \((U_+,\phi_+),(U_-,\Tilde{\phi}_-)\) will be an oriented atlas.

One can show that the real projective space \(\mathbb R P^n\) is orientable if and only if \(n\) is odd or \(n = 0\). More generally, the Grassmannians space \(Gr(k,n)\) is orientable if and only if \(n\) is even or \(n = 1\). The complex projective spaces \(\mathbb C P^n\) and complex Grassmannians \(Gr \mathbb C(k,n)\) are all orientable. This follows because the transition maps for their standard charts, as maps between open subsets of \(\mathbb C^m\), are actually complex-holomorphic, and this implies that as real maps, their Jacobian has positive determinant.

Open subsets

Let \(M\) be a set equipped with an \(m\)-dimensional maximal atlas \(\mathscr{A} = \{(U_\alpha, \phi_\alpha) \}\).

This means that to check that a subset \(U\) is open, it is not actually necessary to verify this condition for all charts. As the above proposition shows, it is enough to check for any collection of charts whose union contains \(U\). In particular, we may take \(\mathscr{A}\) to be any atlas, not necessarily a maximal atlas.

If \(\mathscr{A}\) is an atlas on \(M\), and \(U \subseteq M\) is open, then \(U\) inherits an atlas by restriction: \[\mathscr{A} = \{ (U \cap U_\alpha, \phi_\alpha |_{U \cap U_\alpha } ) \}\]

These properties mean, by definition, that the collection of open subsets of \(M\) define a topology on \(M\). This allows us to adopt various notions from topology:

1. A subset \(A \subseteq M\) is called closed if its complement \(M \setminus A\) is open.

2. \(M\) is called connected if the only subsets \(A \subseteq M\) that are both closed and open are \(A = \emptyset\) and \(A = M\).

3. If \(U\) is an open subset and \(p \in U\), then \(U\) is called an open neighborhood of \(p\). More generally, if \(A \subseteq U\) is a subset contained in \(M\), then \(U\) is called an open neighborhood of \(A\).

The Hausdorff condition in the definition of manifolds can now be restated as the condition that any two distinct points \(p,q \in M\) have disjoint open neighborhoods. (It is not necessary to take them to be domains of coordinate charts.) It is immediate from the definition that domains of coordinate charts are open. Indeed, this gives an alternative way of defining the open sets.

Compact subsets

Another important concept from topology that we will need is the notion of compactness.

In short, \(A \subseteq M\) is compact if every open cover admits a finite subcover.

The proposition is useful, since we can check compactness of \(\phi(A)\) by using the Heine-Borel criterion. For more general subsets of \(M\), we can often decide compactness by combining this result with the following:

A simpler way of verifying compactness is by showing that they are closed and bounded subsets of \(\mathbb R^N\) for a suitable \(N\).

Smooth Maps

Smooth functions on manifolds

The notion of smooth functions on open subsets of Euclidean spaces carries over to manifolds: A function is smooth if its expression in local coordinates is smooth,

Since transition maps are diffeomorphisms, it suffices to check the condition for the charts from any given atlas which need not be the maximal atlas.

Given an open subset \(U \subseteq M\), we say that a function \(f\) is smooth on \(U\) if its restriction \(f |_U\) is smooth. (Here we are using that \(U\) itself is a manifold.) Given \(p \in M\), we say that \(f\) is smooth at \(p\) if it is smooth on some open neighborhood of \(p\).

From the properties of smooth functions on \(\mathbb R^m\), one immediately gets the following properties of smooth functions on manifolds \(M\):

1. If \(f,g \in C^{\infty}(M)\) and \(\lambda, \mu \in \mathbb R\), then \(\lambda f + \mu g \in C^\infty(M)\).

2. If \(f,g \in C^\infty(M)\), then \(f g \in C^\infty(M)\).

3. \(1 \in C^{\infty}(M)\) (where \(1\) denotes the constant function \(p \mapsto 1\))

These properties say that \(C^{\infty}(M)\) is an algebra with unit 1.

Smooth maps between manifolds

The condition for smoothness at p does not depend on the choice of charts. To check smoothness of \(F\), it suffices to take any atlas of M with the property that \(F(U_\alpha) \subseteq V_\alpha\) and then check smoothness of the maps. Smooth maps \(M \rightarrow R\) are the same thing as smooth functions on \(M\), \(C^{\infty} (M,R) = C^{\infty}(M)\).

Diffeomorphisms of manifolds

In other words, a diffeomorphism of manifolds is a bijection of the underlying sets that identifies the maximal atlases of the manifolds. Manifolds that are diffeomorphic are therefore considered ‘the same manifolds’.

Manifolds that are homeomorphic are considered ‘the same topologically’. Since every smooth map is continuous, every diffeomorphism is a homeomorphism.

It is quite possible for two manifolds to be homeomorphic but not diffeomorphic, these are known as exotic manifolds. An exotic sphere is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. It is known that there are no exotic manifold structures on \(\mathbb R^n\) for \(\mathbb R^n\) with \(n \neq 4\), where there are uncountably many such.

Examples of smooth maps

Products, diagonal maps

1. If \(M,N\) are manifolds, then the projection maps are smooth. Take product charts \(U_\alpha \times V_\beta\). \[p^{T}_M : M \times N \rightarrow M, \ \ p^{T}_N : M \times N \rightarrow N\]

2. The diagonal inclusion is smooth. In a coordinate chart around a point, the map is the restriction to a subset of the diagonal inclusion. \[\Delta_M : M \rightarrow M \times M\]

3. Suppose \(F : M \rightarrow N\) and \(F' : M' \rightarrow N'\) are smooth maps. Then the direct product is smooth. \[F \times F' : M \times M' \rightarrow N \times N'\]

The diffeomorphism \(\mathbb R P^1 \cong S^1\)

We have seen that \(\mathbb R P^1 \cong S^1\). To obtain a diffeomorphism, we construct a bijection between the standard atlases of both of the manifolds described previously

The diffeomorphism \(\mathbb C P^1 \cong S^2\)

By a similar reasoning, we find \(\mathbb C P^1 \cong S^2\). For \(S^2\) we use the atlas given by stereographic projection. Regarding \(u\) as a complex number the normis just the absolute value of \(u\), and the transition map becomes \(u \mapsto \frac{1}{u}\). Note that it is not quite the same as the transition map for the standard atlas of \(\mathbb C P^1\), which is given by \(u \mapsto u^{-1}\). We obtain a unique diffeomorphism such that \(\phi_+ \circ F \circ \phi_0^{-1}\) is the identity and \(\phi_- \circ F \circ \phi_1^{-1}\) is complex conjugation

Maps to and from projective space

The quotient map \(\pi\) is smooth, as one verifies by checking in the standard atlas for \(\mathbb R P^n\). \[\pi : \mathbb R^{n+1} \setminus \{0\} \rightarrow \mathbb R P^N, \ \ x = (x^0, \dots, x^n) \mapsto (x^0, \dots, x^n)\] Given a map \(F : \mathbb R P^n \rightarrow N\) to a manifold \(N\), let \(\Tilde{F} = F \circ \pi : \mathbb R ^{n+1}\setminus \{0\} \rightarrow N\) be its composition with the projection map \(\pi : \mathbb R ^{n+1}\setminus \{0\} \rightarrow \mathbb R P^n\). That is, \(\Tilde{F} (x^0 , \dots, x^n ) = F(x^0 : \dots : x^n )\).

We claim that the map \(F\) is smooth if and only the corresponding map \(\Tilde{F}\) is smooth. One direction is clear: If \(F\) is smooth, then \(\Tilde{F} = F \circ \pi\) is a composition of smooth maps. For the other direction, assuming that \(\Tilde{F}\) is smooth, note that for the standard chart \((U_j ,\phi_j)\), and the maps \[(F \circ \phi^{-1}_j )(u^1 ,\dots, u^n ) = \Tilde{F}(u^1 ,\dots, u^i ,1,u ^{i+1} ,\dots,u^n ),\] are smooth. An analogous argument applies to the complex projective space \(\mathbb C P^n\) , taking the \(x^i\) to be complex numbers \(z^i\)

The Hopf fibration, a.k.a. the quotient map \(S^{2n+1} \rightarrow \mathbb C P^n\)

As mentioned above, quotient map \(q : C^{n+1} \setminus \{0\} \rightarrow \mathbb C P^n\) is smooth. Since any class \([z] = (z 0 : \dots : z^n )\) has a representative with \(|z^0|^2 + \dots +|z^n|^2 = 1\), and \(|z^i|^2 = (x^i)^2 + (y^i)^2\) for \(z^i = x^i + \sqrt{-1}y^i\), we may also regard \(\mathbb C P^n\) as a set of equivalence classes in the unit sphere \(S^{2n+1} \subseteq \mathbb R^{2n+2} = \mathbb C^{n+1}\). The resulting quotient map \[\pi : S^{2n+1} \rightarrow \mathbb C P^n\] is again smooth, because it can be written as a composition of two smooth maps \(\pi = q \circ \tau\) where \(\tau : S ^{2n+1} \mapsto \mathbb R^{2n+2} \setminus \{0\} = \mathbb C^{n+1} \setminus \{0\}\) is the inclusion map.

For any \(p \in \mathbb C P^n\) , the corresponding fiber \(\pi^{-1} (p) \subset S^{2n+1}\) is diffeomorphic to a circle \(S^1\) (which we may regard as complex numbers of absolute value 1). Indeed, given any point \((z^0 , \dots ,z^n ) \in \pi^{-1} (p)\) in the fiber, the other points are obtained as \((\lambda z^0 ,\dots, \lambda z^n )\) where \(|\lambda| = 1\).

In other words, we can think of \[S^{2n+1} = \bigcup_{p \in \mathbb C P^n} \pi^{- 1}(p)\] as a union of circles, parametrized by the points of \(\mathbb C P^n\). This is an example of a fiber bundle or fibration.

An import important case occurs when \(n = 1\). Identifying \(\mathbb C P^1 \cong S^2\) as above, the map \(\pi\) becomes a smooth map \(\pi : S^3 \rightarrow S^2\) with fibers diffeomorphic to \(S^1\). This map appears in many contexts; it is called the Hopf fibration.

Let \(S \in S^3\) be the ‘south pole’, and \(N \in S^3\) the ‘north pole’. We have that \(S^3 - \{S\} \cong R^3\) by stereographic projection. The set \(\pi^{-1} (\pi(S)) - \{S\}\) projects to a straight line (think of it as a circle with ‘infinite radius’). The fiber \(\pi ^{-1} (N)\) is a circle that goes around the straight line. If \(Z \subseteq S^2\) is a circle at a given ‘latitude’, then \(\pi^{-1} (Z)\) is is a 2-torus. For \(Z\) close to north pole \(N\) this 2-torus is very thin, while for \(Z\) approaching the south pole \(S\) the radius goes to infinity. Each such 2-torus is itself a union of circles \(\pi^{-1} (p), p \in Z\). Those circles are neither the usual ‘vertical’ or ‘horizontal’ circles of a 2-torus in \(\mathbb R^3\) , but instead are ‘tilted’. In fact, each such circle is a ‘perfect geometric circle’ obtained as the intersection of its 2-torus with a carefully positioned affine 2-plane. Moreover, any two of the circles \(\pi^{-1} (p)\) are ‘linked’ as though they were in a chain.

A calculation shows that over the charts \(U_+,U_-\) from stereographic projection, the Hopf fibration is just a product. That is, one has \[\pi^{-1} (U_+) \cong U_+ \times S^1, \ \ \pi^{-1} (U_-) \cong U_- \times S^1\] In particular, the pre-image of the closed upper hemisphere is a solid 2-torus \(D^2 \times S^1\) (with \(D^2 = \{ z \in \mathbb C| |z| \leq 1 \}\) the unit disk), geometrically depicted as a 2-torus in \(\mathbb R^3\) together with its interior. We hence see that the \(S^3\) may be obtained by gluing two solid 2-tori along their boundaries \(S^1 \times S^1\).

Submanifolds

Strictly speaking, a submanifold chart for \(S\) is not a chart for \(S\), but is a chart for \(M\) which is adapted to \(S\). Submanifold charts restrict to charts for \(S\), and this may be used to construct an atlas for \(S\).

In other words, the topology of \(S\) as a manifold coincides with the ‘subspace topology’ as a subset of the manifold \(M\).

As a consequence, if a manifold \(M\) can be realized realized as a submanifold \(M \subseteq \mathbb R^n\), then \(M\) is compact with respect to its manifold topology if and only if it is compact as a subset of \(R^n\), if and only if it is a closed and bounded subset of \(\mathbb R^n\).

Smooth maps of maximal rank

Let \(F \in C^\infty(M,N)\) be a smooth map. Then the fibers (level sets) \(F^{-1}(q) = \{x \in M| F(x) = q\}\) for \(q \in N\) need not be submanifolds, in general.

The rank of a smooth map

Let \(U \subseteq \mathbb R^m\) and \(V \subseteq \mathbb R^n\) be open subsets, and \(F \in C^\infty(U,V)\) a smooth map.

Equivalently, \(D_p F\) is the \(n \times m\) matrix of partial derivatives and the rank of \(F\) at \(p\) is the rank of this matrix i.e., the number of linearly independent rows or the number of linearly independent columns.

Local diffeomorphisms

The theorem tells us that for a smooth bijection, a sufficient condition for smoothness of the inverse map is that the differential (i.e., the first derivative) is invertible everywhere.

A smooth map \(F \in C^\infty(M,N)\) is called a local diffeomorphism if \(\text{dim} M = \text{dim}N\), and \(F\) has maximal rank everywhere. By the theorem, this is equivalent to the condition that every point \(p\) has an open neighborhood \(U\) such that \(F\) restricts to a diffeomorphism \(U \rightarrow F(U)\).

Level sets, submersions

Again, this result has a version for manifolds:

Note that regular values are only possible if \(\text{dim}N \leq \text{dim}M\). Note also that all points of \(N\) that are not in the image of the map F are considered regular values. We may restate the theorem as follows:

Example: The Steiner surface

[Omitted]

Immersions

The manifolds version reads as follows:

The Tangent Bundle

For embedded submanifolds \(M \subseteq \mathbb R^n\), the tangent space \(T_p M\) at \(p \in M\) can be defined as the set of all velocity vectors \(v = \gamma(0\)), where \(\gamma : J \rightarrow M\) is a smooth curve with \(\gamma(0) = p\); here \(J \subseteq R\) is an open interval around \(0\). It turns out that \(T_pM\) becomes a vector subspace of \(\mathbb R^n\). For a general manifold, we will define \(T_pM\) as a set of directional derivatives.

The following local coordinate description makes it clear that \(T_pM\) is a linear subspace of the vector space \(L(C^\infty(M),R)\) of linear maps \(C^\infty(M) \rightarrow R\), of dimension equal to the dimension of \(M\).

We can use this result as an alternative definition of the tangent space, namely:

It is not immediately obvious from this second definition that \(T_pM\) is independent of the choice of coordinate chart, but this follows from the equivalence with the first definition. Any choice of coordinate chart \((U,\phi)\) around \(p\) defines a vector space isomorphism \(T_pM \cong \mathbb R^m\), taking \(v\) to \(a = (a^1 ,\dots,a^m)\). In particular, we see that if \(U \subseteq \mathbb R^m\) is an open subset, and \(p \in U\), then \(T_pU\) is the subspace of the space of linear maps \(C^\infty(M) \rightarrow \mathbb R\) spanned by the partial derivatives at \(p\).

We now describe yet another approach to tangent spaces which again characterizes “directional derivatives” in a coordinate-free way, but without reference to curves \(\gamma\). Note first that every tangent vector satisfies the product rule, also called the Leibniz rule:

Alternatively, in local coordinates it is just the product rule for partial derivatives. It turns out that the product rule completely characterizes tangent vector:

Tangent map

Definition of the tangent map, basic properties

The following definition generalizes the derivative to smooth maps between manifolds.

Coordinate description of the tangent map

Now that we have recognized \(TpF\) as the derivative expressed in a coordinate-free way, we may liberate some of our earlier definitions from coordinates:

• The rank of \(F\) at \(p \in M\), denoted \(\text{rank}p(F)\), is the rank of the linear map \(T_pF\).

• \(F\) has maximal rank at \(p\) if \(\text{rank}p(F) = min(dim M, dim N)\).

• \(F\) is a submersion if \(T_pF\) is surjective for all \(p \in M\),

• \(F\) is an immersion if \(T_pF\) is injective for all \(p \in M\),

• \(F\) is a local diffeomorphism if \(T_pF\) is an isomorphism for all \(p \in M\).

• \(p \in M\) is a critical point of \(F\) is \(T_pF\) does not have maximal rank at \(p\).

• \(q \in N\) is a regular value of \(F\) if \(T_pF\) is surjective for all \(p \in F^{-1} (q)\) (in particular, if \(q \not\in F(M))\)

• \(q \in N\) is a singular value if it is not a regular value.

Tangent spaces of submanifolds

Suppose \(S \subseteq M\) is a submanifold, and \(p \in S\). Then the tangent space \(T_pS\) is canonically identified as a subspace of \(T_pM\). Indeed, since the inclusion \(i : S \mapsto M\) is an immersion, the tangent map is an injective linear map, \(T_pi : T_pS \rightarrow T_pM\), and we identify \(T_pS\) with the subspace given as the image of this map.

Recall, the kernel of a linear mapping, also known as the null space or nullspace, is the set of vectors in the domain of the mapping which are mapped to the zero vector.

Example: Steiner’s surface revisited

[Omitted]

The tangent bundle

Vector Fields

Vector fields as derivations

Since each \(X_p\) satisfy the product rule (at \(p\)), it follows that \(X\) itself satisfies a product rule. We can use this as an alternative definition:

We can also express the smoothness of the tangent vectors \(X_p\) in terms of coordinate charts \((U,\phi)\). Recall that for any \(p \in U\), and all \(f \in C^\infty(M)\), the tangent vector \(X_p\) is expressed as \[X_p(f) = \sum_{i=1}^m a^i (u) \frac{\partial}{\partial u^i} \Bigr |_{u=\phi(p)} (f \circ \phi^{-1})\]

Vector fields as sections of the tangent bundle

It is common practice to use the same symbol X both as a linear map from smooth functions to smooth functions, i.e. \(X : M \rightarrow TM\), or as a map into the tangent bundle, \(X : C^\infty(M) \rightarrow C^\infty(M)\). The latter case can also be expressed as the ‘Lie derivative’ to avoid confusion: \(L_X : C^\infty(M) \rightarrow C^\infty(M)\)

Lie brackets

Note: When calculating Lie brackets \(X \circ Y - Y \circ X\) of vector fields \(X,Y\) in local coordinates, it is not necessary to work out the second order derivatives – we know in advance that these are going to cancel out.

Let \(S \subseteq M\) be a submanifold. A vector field \(X \in \mathfrak X(M)\) is called tangent to \(S\) if for all \(p \in S\), the tangent vector \(X_p\) lies in \(T_pS \subseteq T_pM\). (Thus \(X\) restricts to a vector field \(X|_S \in \mathfrak X(S)\).)

Related vector fields

The \(F\)-relation of vector fields also has a simple interpretation in terms of the ‘differential operator’ picture.

Flows of vector fields

Recall, For any curve \(\gamma : J \rightarrow M\), with \(J \subseteq \mathbb R\) an open interval, and any \(t \in J\), the velocity vector \(\dot{\gamma}(t) \equiv \frac{d\gamma}{dt} \in T_{\gamma(t)}M\) is defined as the tangent vector, given in terms of its action on functions as \((\dot{\gamma}(t))(f) = \frac{d}{dt} f(\gamma(t))\). (The dot signifies a t-derivative.)

Equivalently, one may think of the velocity vector as the image of \(\frac{\partial}{\partial t} |_{t} \in T_tJ \cong \mathbb R\) under the tangent map \(T_t \gamma : \dot{\gamma}(t) = (T_t\gamma)( \frac{\partial}{\partial t}|_t)\).

Geometrically, this means that at any given time \(t\), the value of \(X\) at \(\gamma(t)\) agrees with the velocity vector to \(\gamma\) at \(t\), i.e. \(\frac{\partial}{\partial t} \sim_\gamma X\).

Thus, \(J_{x_0}\) is the maximal open interval on which the solution is defined.

For a general vector field \(X \in \mathfrak X(M)\) on manifolds, Equation \(\dot{\gamma}(t) = X_{\gamma(t)}\) becomes \(\frac{dx^i}{dt} = a^i (x(t)), i = 1,\dots,m\) after introduction of local coordinates. The existence and uniqueness theorem for ODE’s extends to manifolds, as follows:

Note that the uniqueness part uses the Hausdorff property in the definition of manifolds. Indeed, the uniqueness part may fail for non-Hausdorff manifolds.

Intuitively, \(\Phi_t(p)\) is obtained from the initial point \(p \in M\) by flowing for time \(t\) along the vector field \(X\). One expects that first flowing for time \(t\), and then flowing for time \(s\), should be the same as flowing for time \(t +s\). Indeed one has the following flow property

We see in particular that for any \(t\), the map \(\Phi_t : U_t \rightarrow M\) is a diffeomorphism onto its image \(\Phi_t(U_t) = U_{-t}\), with inverse \(\Phi_{-t}\). Let \(X\) be a vector field, and \(\mathscr J = \mathscr J^X\) be the domain of definition for the flow \(\Phi = \Phi^X\) .

A vector field may fail to be complete if a solution curve escapes to infinity in finite time. This suggests that a vector fields \(X\) that vanishes outside a compact set must be complete, because the solution curves are ‘trapped’ and cannot escape to infinity:

In short, vector fields are \(F\)-related if and only if their flows are \(F\)-related. \(\Phi^*_t : C^\infty (M) \rightarrow C^\infty (M), \Phi^* t : \mathfrak X(M) \rightarrow \mathfrak X(M)\).

Geometric interpretation of the Lie bracket

For any smooth map \(F \in C^\infty(M,N)\) we defined the pull-back \[F^* : C ^\infty (N) \rightarrow C ^\infty (M), \ g \mapsto g \circ F.\] If \(F\) is a diffeomorphism, then we can also pull back vector fields: \(F^* : X(N) \rightarrow X(M), \ Y \mapsto F^*Y\), by the condition \((F^*Y)(F^* g) = F^* (Y(g))\) for all functions \(g\). That is, \(F^* Y \sim_F Y\), or in more detail \((F^*Y)_p = (T_pF)^{ -1}Y_{F(p)}\). By Theorem 5.2, we have \(F^*[X,Y] = [F^*X,F^*Y]\).

Any complete vector field \(X \in \mathfrak X(M)\) with flow \(\Phi_t\) gives rise to a families of pull-back map. \[\Phi^*_t : C ^\infty (M) \rightarrow C^\infty (M), \ \Phi^*_t : X(M) \rightarrow X(M)\]

Thus, the Lie bracket \([X,Y]\) measures ‘infinitesimally’ how the vector field \(Y\) changes along the flow of \(X\). Note that in particular, \(L_XY\) is skew-symmetric in \(X\) and \(Y\) – this is not obvious from the definition. One can also interpret the Lie bracket as measuring how the flows of \(X\) and \(Y\) fail to commute.

Frobenius theorem

We saw that for any vector field \(X \in \mathfrak X(M)\), there are solution curves through any given point \(p \in M\). The image of this curve is an (immersed) submanifold to which \(X\) is everywhere tangent. One might similarly ‘integral surfaces’ for pairs of vector fields, and ‘integral submanifolds’ for collections of vector fields

Thus, for any involutive subbundle \(E \subseteq TM\), then any \(p \in M\) has an open neighborhood \(U\) with a nice decomposition into \(r\)-dimensional submanifolds. One calls such a decomposition (or sometimes the involutive subbundle E itself) a (local) foliation. A foliation gives a decomposition into submanifolds on a neighborhood of any given point. Globally, the integral submanifolds are often only immersed submanifolds, given by immersions \(i : S \rightarrow M\) with \((T_pi)(T_pS) = E_p\) for all \(p \in S\).

Differential Forms

Review: Differential forms on \(\mathbb R^m\)

Differential forms are an approach to solving multivariable calculus problems that is independent of coordinates. They provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.

Let \(\Omega^k (U)\) be the vector space consisting of such expressions, with pointwise addition. It is convenient to introduce a short hand notation \(I = {i_1,\dots,i_k}\) for the index set, and write \(\omega = \sum_I \omega_I dx^I\) with \(\omega_I = \omega_{i_1 \dots i_k}\), and \(dx^I = dx^{i1} \wedge \dots \wedge dx^{ik}\).

Since a \(k\)-form is determined by these functions \(\omega_I\), and since there are \(\frac{m!}{k!(m-k)!}\) ways of picking \(k\)-element subsets from \(\{1,\dots,m\}\), the space \(\Omega^k (U)\) can be identified with vector-valued smooth functions, \(\Omega^k (U) = C^\infty (U, \mathbb R ^{\frac{m!}{k!(m-k)!}})\).

An associative product operation \(\Omega^k (U) \times \Omega^l (U) \rightarrow \Omega^{k+l} (U)\) by the ‘rule of computation’ \(dx^i \wedge d x^j = -dx^j \wedge d x^i\) for all \(i, j\); in particular \(dx^i \wedge dx^i = 0\).

The key property of the exterior differential is the following fact:

Note that we can regard \(\omega\) as a form on all of \(\mathbb R^m\), due to the compact support condition.

Our aim is now to define differential forms on manifolds, beginning with 1-forms. Even though 1-forms on \(U \subseteq \mathbb R^m\) are identified with functions \(U \rightarrow \mathbb R^m\), they should not be regarded as vector fields, since their transformation properties under coordinate changes are different. In fact, while vector fields are sections of the tangent bundle, the 1-forms are sections of its dual space, the cotangent bundle. We will thus begin with a review of dual spaces in general.

Dual spaces

If \(E\) is finite-dimensional, then the dual space is also finite-dimensional, and \(\dim E^* = \dim E\). It is common to write the value of \(\alpha \in E^*\) on \(v \in E\) as a pairing, using the bracket notation \(\langle \alpha, v \rangle := \alpha(v);\). (In physics, it is common to use Dirac bra-ket notation \(\langle \alpha | v \rangle := \alpha(v)\).)

The element \(\alpha \in E^*\) is described in terms of the dual bases as \(\alpha = \sum_{j=1}^r \alpha_j e^j, \ \alpha_j = \langle \alpha, e_j \rangle\). Similarly, for vectors \(v \in E\) we have \(v = \sum_{i=1}^r v^i e_i, \ v^i = \langle e^i , v \rangle\).

This satisfies \((R^*)^* = R\), and under the composition of linear maps, \((R_1 \circ R_2)^* = R^*_2 \circ R^*_1\). In terms of basis \(e_1,\dots, e_r\) of \(E\) and \(f_1,\dots, f_s\) of \(F\), and the corresponding dual bases (with upper indices), a linear map \(R : E \rightarrow F\) is given by the matrix with entries \(R_i^j = \langle f^j , R(e_i)\rangle\), while \(R^*\) is described by the transpose of this matrix (the roles of \(i\) and \(j\) are reversed). Thus, \((R^*)^j_i = R_i^j\).

Cotangent spaces

Thus, a co(tangent) vector at \(p\) is a linear functional on the tangent space, assigning to each tangent vector at \(p\) a number. The very definition of the tangent space suggests one such functional: Every function \(f \in C^\infty(M)\) defines a linear map, \(T_pM \rightarrow \mathbb R, v \mapsto v(f)\). This linear functional is denoted \((d f)_p \in T^*_p M\).

1-forms

Similar to the definition of vector fields, one can define co-vector fields, more commonly known as 1-forms: Collections of covectors \(\alpha_p \in T^*_p M\) depending smoothly on the base point.

Let us verify that a 1-form can be regarded as a collection of covectors:

The first example of a 1-form is described in the following definition.

Pull-backs of function and 1-forms

Recall that for any manifold \(M\), the vector space \(C^\infty(M)\) of smooth functions is an algebra, with product the pointwise multiplication. Any smooth map \(F : M \rightarrow M'\) between manifolds defined an algebra homomorphism, called the pull-back \[F^* : C^\infty (M') \rightarrow C^\infty (M), \ \ f \mapsto F ^* (f) := f \circ F.\] The fact that this preserves products is the following simple calculation: \[(F^* (f)F^* (g))(p) = f(F(p))g(F(p)) = (f g)(F(p)) = F^* (f g)(p).\] Given another smooth map \(F' : M' \rightarrow M''\) we have \((F'\circ F)^* \circ F^* \circ (F' )^*\).

Let \(F \in C^\infty(M,N)\) be a smooth map. Recall that for vector fields, there is no general ‘push-forward’ or ‘pull-back’ operation, unless \(F\) is a diffeomorphism. For 1-forms the situation is better: for any \(p \in M\) one has the dual to the tangent map \[T^*_p F = (T_pF)^* : T^*_{F(p)}N \rightarrow T^*_p M.\] For a 1-form \(\beta \in \Omega^1 (N)\), we can therefore define \((F^* \beta)_p := (T^*_p F)(\beta_{F(p)})\)

The Lemma shows that we have a well-defined pull-back map \(F^* : \Omega^1 (N) \rightarrow \Omega^1 (M), \beta \mapsto F^* \beta\). Under composition of two maps, \((F_1 \circ F_2)^* = F^*_2 \circ F^*_1\). The pull-back of forms is related to the pull-back of functions, \(g \mapsto F^*g = g \circ F\):

Recall once again that while \(F \in C^\infty(M,N)\) induces a tangent map there is no natural push-forward operation for vector fields. By contrast, for cotangent bundles there is no naturally induced map from \(T^*N\) to \(T^*M\) (or the other way), yet there is a natural pull-back operation for 1-forms. For any related vector fields \(X \sim_F Y\), and \(\beta \in \Omega^1 (N)\), we then have that \((F^* \beta)(X) = F^* (\beta(Y))\). Indeed, at any given \(p \in M\) this just becomes the definition of the pullback map.

Integration of 1-forms

Given a curve \(\gamma : J \rightarrow M\) in a manifold, and any 1-form \(\alpha \in \Omega^1 (M)\), we can consider the pull-back \(\gamma^*\alpha \in \Omega^1 (J)\). By the description of 1-forms on \(\mathbb R\), this is of the form \(\gamma^*\alpha = f(t)dt\) for a smooth function \(f \in C^\infty(J)\).

To discuss integration, it is convenient to work with closed intervals rather than open intervals. Let \([a,b] \subseteq R\) be a closed interval. A map \(\gamma : [a,b] \rightarrow M\) into a manifold will be called smooth if it extends to a smooth map from an open interval containing \([a,b]\). We will call such a map a smooth path.

The fundamental theorem of calculus has the following consequence for manifolds. It is a special case of Stokes’ theorem

2-forms

k-forms

Definition

Here, skew-symmetry means that \(\alpha(X_1,\dots, X_k)\) changes sign under exchange of any two of its elements. The \(C^\infty(M)\)-multilinearity means \(C^\infty(M)\)-linearity in each argument, similar to the condition for 2-forms. It implies \(\alpha\) is local in the sense that the value of \(\alpha(X_1,\dots,X_k)\) at any given \(p \in M\) depends only on the values \(X_1|p,\dots,X_k |p \in T_pM\).

If \(\alpha_1,\dots,\alpha_k\) are 1-forms, then one obtains a k-form \(\alpha =: \alpha_1\wedge\dots\wedge\alpha_k\) by wedge product.

Using \(C^\infty\)-multilinearity, a k-form on \(U \subseteq R\) m is uniquely determined by its values on coordinate vector fields. i.e. by the functions, \[\alpha_{i_1\dots i_k} = \alpha \bigg ( \frac{\partial}{\partial x^{i_1}}, \dots , \frac{\partial}{ \partial x ^{i_k}} \bigg )\] Moreover, by skew-symmetry we only need to consider ordered index sets \(I = {i_1,\dots,i_k} \subseteq {1,\dots,m}\), that is, \(i_1 < \dots < i_k\). Using the wedge product notation, we obtain \[\alpha = \sum_{i_1<\dots<i_k} \alpha_{i_1\dotsi_k} dx^{i_1} \wedge \dots dx^{i_k}.\]

Wedge product

The wedge product is graded commutative: If \(\alpha \in \Omega^k (M)\) and \(\beta \in \Omega^l (M)\) then \(\alpha \wedge \beta = (-1) ^{kl}\beta \wedge \alpha\). Furthermore, it is associative:

Exterior differential

Recall that we defined the exterior differential on functions by the formula \((d f)(X) = X(f)\). We will now extend this definition to all forms.

Since \(d\circ d = 0\), the exact \(k\)-forms are a subspace of the space of closed \(k\)-forms; a necessary condition for \(\alpha\) to be exact is that it is closed.

Lie derivatives and contractions

Another important operator on forms is the Lie derivative:

These operators, \(d, L_X, \iota_X\), have the following compatibilities with the wedge product: For \(\alpha \in \Omega^k (M)\) and \(\beta \in \Omega^l (M)\) one has \[\begin{aligned} d(\alpha \wedge\beta) &= (d\alpha)\wedge\beta + (-1) k\alpha \wedge d\beta,\\ L_X (\alpha \wedge\beta) &= (L_X\alpha)\wedge\beta +\alpha \wedge L_X \beta,\\ \iota_X (\alpha \wedge\beta) &= (\iota_X\alpha)\wedge\beta + (-1) k\alpha \wedge\iota_X \beta.\end{aligned}\]

One says that \(L_X\) is an even derivation relative to the wedge product, whereas \(d,\iota_X\) are odd derivations. They also satisfy important relations among each other: \[\begin{aligned} d \circ d = 0 \\ L_X \circ L_Y -L_Y \circ L_X = L_{[X,Y]} \\ \iota_X \circ \iota_Y +\iota_Y \circ \iota_X = 0 \\ d \circ L_X -L_X \circ d = 0 \\ L_X \circ \iota_Y -\iota_Y \circ L_X = \iota_[X,Y] \\ \iota_X \circ d+d \circ \iota_X = L_X .\end{aligned}\] This collection of identities is referred to as the Cartan calculus, , and in particular the last identity is called the Cartan formula.

Pull-backs

The pull-back map satisfies \(d(F^*\beta) = F^*d\beta\), and for a wedge product of forms, \(F^* (\beta_1 \wedge \beta_2) = F^* \beta_1 \wedge F^* \beta_2\).

The Lie derivative \(L_X \alpha\) of a differential form with respect to a vector field \(X\) has an important interpretation in terms of the flow \(\Phi_t\) of \(X\). Assuming for simplicity that \(X\) is complete (so that \(\Phi_t\) is a globally defined diffeomorphism), one has the formula \[L_X\alpha = \frac{d}{dt} \Bigr |_{t=0} \Phi^*_t \alpha.\] The formula shows that \(L_X\) measures to what extent \(\alpha\) is invariant under the flow of \(X\).

Integration of differential forms

Differential forms of top degree can be integrated over oriented manifolds. Let \(M\) be an oriented manifold of dimension \(m\), and \(\omega \in \Omega^m(M)\). Let \(supp(\omega)\) be the support of \(\omega\). If \(supp(\omega)\) is contained in an oriented coordinate chart \((U,\phi)\), then one defines \[\int_M \omega = \int_{\mathbb R^m} f(x)dx^1\cdots dx^m\]

where \(f \in C^\infty(\mathbb R^m)\) is the function, with \(supp(f) \subseteq \phi(U)\), determined from \[(\phi^{-1} )^* \omega = f dx^1 \wedge \cdots \wedge dx^m.\] This definition does not depend on the choice of oriented chart.

If \(\omega\) is not necessarily supported in a single oriented chart, we proceed as follows. Let \((U_i , \phi^i), \ i = 1,\dots,r\) be a finite collection of oriented charts covering \(supp(\omega)\). Together with \(U_0 = M \setminus supp(\omega)\) this is an open cover of \(M\).

Indeed, partitions of unity exists for any open cover, not only finite ones. Let \(\chi_0, \dots, \chi_r\) be a partition of unity subordinate to this cover. We define \[\int_M \omega = \sum_{i=1}^r \int_M \chi_i \omega\] where the summands are defined as above, since \(\chi_i\omega\) is supported in \(U_i\) for \(i \geq 1\). It can be shown that this is well defined, independent of the choice of oriented coordinate charts.

Integration over oriented submanifolds

Let \(M\) be a manifold, not necessarily oriented, and \(S\) is a k-dimensional oriented submanifold, with inclusion \(i : S \rightarrow M\). We define the integral over \(S\), of any k-form \(\omega \in \Omega^k (M)\) such that \(S \cap supp(\omega)\) is compact, as follows:

\[\int_S \omega = \int_S i^* \omega.\] Of course, this definition works equally well for any smooth map from \(S\) into \(M\). For example, the integral of compactly supported 1-forms along arbitrary paths \(\gamma : \mathbb R \rightarrow M\) is defined. Note also that \(M\) itself does not have to be oriented, it suffices that \(S\) is oriented.

Stokes’ theorem

Let \(M\) be an \(m\)-dimensional oriented manifold.

Recall that we are considering \(D\) inside an oriented manifold \(M\). The boundary \(\partial D\) may be covered by oriented submanifold charts \((U,\phi)\), in such a way that \(\partial D\) is given in the chart by the condition \(x^1 = 0\), and \(D\) by the condition \(x^1 \leq 0\): \[\phi (U \cap D) = \phi (U) \cap \{x \in \mathbb R^m | x ^1 \leq 0 \}.\]

We call oriented submanifold charts of this kind ‘region charts’.

In particular, \(\partial D\) is again an oriented manifold. To repeat: If \(x^1 ,\dots, x^m\) are local coordinates near \(p \in \partial D\), compatible with the orientation and such that \(D\) lies on the side \(x^1 \leq 0\), then \(x^2 ,\dots, x^m\) are local coordinates on \(\partial D\). This convention of ‘induced orientation’ is arranged in such a way that the Stokes’ theorem holds without extra signs.

For an \(m\)-form \(\omega\) such that \(supp(\omega)\cap D\) is compact, the integral \(\int_D \omega\) is defined similar to the case of \(D = M\).

As explained above, the right hand side means \(\int_{\partial D} i^* \alpha\), where \(i : \partial D \rightarrow M\) is the inclusion map.

Note that it does not suffice that \(d\alpha\) has compact support. A typical application of Stokes’ theorem shows that for a closed form \(\omega \in \Omega^k (M)\), the integral of \(\omega\) over an oriented compact submanifold does not change with smooth deformations of the submanifold.

If \(F_t\) is an embedding, then this is the integral of \(\omega\) over the submanifold \(F_t(S) \subseteq M\).

The previous theorem shows that if \(S\) is oriented, and if there is a closed form \(\omega \in \Omega^k (M)\) with \[\int_S \phi^* \omega \neq \int_S (\phi' )^* \omega\] then \(\phi\) cannot be smoothly deformed into \(\phi 0\).

Volume forms

Note that volume forms are always closed, for degree reasons (since \(\Omega^{m+1} (M) = 0\)). But on a compact manifold, they cannot be exact:

De Rham Cohomology

The exterior derivative converts the algebra of differential forms on a manifold into a graded differential algebra. The corresponding cohomology is called the de Rham cohomology algebra.

See Text for full details:

http://im0.p.lodz.pl/ kubarski/AnalizaIV/Wyklady/GHV/ITOM/G-H-V-1

In article 1 it is shown that the de Rham cohomology satisfies the dimension, homotopy, disjoint union, and Mayer-Vietoris axioms. In article 2 various examples (retracts, PoincarC lemma, cohomology of Sn, and RP") are discussed. In article 3 everything is done again (with the appropriate modifications) for differential forms with compact carrier. In article 4 the integral is used to establish the PoincarC duality theorem for a smooth orientable manifold. This theorem is applied in article 5 (sec. 5.13 and 5.14) to determine the nth de Rham cohomology space for any n-manifold (orientable or nonorientable). In sec. 5.15 the duality theorem is used to show that a compact manifold has finitedimensional de Rham cohomology. The de Rham cohomology of the product of two manifolds is computed in article 6 (Kunneth theorems). In article 7 one version of the de Rham theorem is established. The results of this article are not quoted elsewhere in the book.

Overview of Riemannian Geometry

Note, there may be changes in notation from switching reading source.

On \(\mathbb R^n\) we have notions like:

1. Length of vectors angles orthonormal/orthogonal geometry

2. Areas volumes length of curves

3. Distance between 2 points

4. Parallel transport

Observe: All of the elements of Euclidean geometry on \(\mathbb R\) is entirely encoded in the inner product. We introduce geometry into smooth manifolds by making a choice of inner products on each \(TpM\) that varies smoothly from point to point. This is an inner product.

The manifold has the following properties:

1. The length of vectors for \(v \in T_pm\), \(||v|| = \sqrt{g(v,v)}\), \(\cos (angle v w) = \frac{g(v,w)}{||v||||w||}\).

   Lemma 125. Near each point, there exists an orthonormal local frame.

2. The length of curve \(\gamma := \int_a^b \sqrt{g(\gamma(t),\gamma ' (t))} dt\).

3. Distance \(d: M \times M \rightarrow [0, \infty)\) is defined by \(d(p,q) = \inf(\int_0^1 || \gamma '(t)|| )dt\) where \(\gamma(0) = p, \gamma(1) = q\).

   This makes \((M,d)\) a metric space so that the metric topology is the same as the original topology. One can show that \((M,d)\) is complete as a metric space iff \(d(p,q)\) is obtained by a curve \(\gamma\) for all \(p,q \in M\).

4. There is a natural isomorphism \(\Phi_p: T_pM \rightarrow T^*_pM, \ v \mapsto (v^* : w \mapsto g(v,w))\). So there is a module isomorphism \(\Phi : \mathfrak X \rightarrow \Omega '(M)\).

5. Let \(S \subseteq M\) be a \(k\)-dimensional manifold. Then \(i^*g\) (induced metric on S) is a Riemannian metric on \(S\) giving \(S\) the unique geometry inherited by \(M\).

6. Suppose \(M\) in oriented. There does not exist a nowhere vanishing n-form called the volume form satisfying \(\omega(x_1,\dots, x_n) = 1\) whenever \(x\) is a local orthonormal frame that has positive orientation. This allows for integration of functions for \(f \in C^\infty(M)\) defined as \(\int_M f := \int_m f \omega\). Also, if \(M\) is compact, we have \(vol(M) = \int_M 1 \ \int_M \omega\). Similarly, \(Vol_{k-dim}(S) = \int_S 1\) with respect to the induced metric \(i^* g\) on \(S\).

7. We say \((M, g)\), \((\Tilde{M}, \Tilde{g})\) are isometric if there exists a diffeomorphism \(F: M \rightarrow \Tilde{M}\) such that \(g = F^*\Tilde{g}\)

Covariant derivative and curvature

If a parallel transport of a vector along a curve has the property that \(D_{\gamma'(t)} = 0\), then \(\gamma\) is the shortest distance between \(p\) and \(q\) so \(\gamma\) is a line.

On a sphere in \(S^2\), parrallel transport is equivalent to \((\delta _{\gamma'(t)} X )^T = 0\) intrinsic to the round metric \((S^3, g)\). Then the shortest paths are the curves which form great circles.

Define \((\Tilde{\delta _{X}} Y)= (\delta_X Y)^T\) as the connection operator for \(X,Y\in \mathfrak{X}(S^2)\). This operator is intrinsic to \(S^2\) and satisfies:

1. \(C^\infty\)-linear w.r.t. \(X\).

2. \(\mathbb R\)-linear w.r.t. \(Y\).

3. \((\Tilde{\delta _{X}} fY) = X(f)Y + f\Tilde{\delta_X}Y\).

4. \(\Tilde{\delta_x}y - \Tilde{\delta_y X} = L_x Y\)

5. \(Z(g(x,y)) - g(\Tilde{\delta_Z} X,Y) - g(x, \Tilde{\delta_Z}y) = 0\)

Let \(\gamma\) be a curve, then \[\frac{d}{dt} (g(X_\gamma(t)), Y_\gamma (T)) = g(\Tilde{\gamma'(t)}X, Y ) + g(X, \Tilde{\gamma'(t)} Y).\]

Note, the converse is not true. A counter example of a great circle minus a small segment will not be the shortest distance between the two points.

Let \((U, \phi)\) be a chart. \[\delta_{\frac{\partial}{\partial x^i}} \frac{\partial }{\partial x^i} = \sum_k \Gamma_{ij}^k \frac{\partial}{\partial x^k}\] Where \(\Gamma_{ij}^k\) are the Levi-Civita connection coeffecients and are completely determined by the metric. \[\Gamma _{ij}^{k}={\tfrac {1}{2}}g^{mk}\left( \frac{\partial g_{im}}{\partial x^j} + \frac{ \partial g_{jm}}{\partial x^i} -\frac{partial g_{ij}}{\partial x ^m}\right)\]

\[g_{ij} = g(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j})\]

The term covariant derivative is often used for the Levi-Civita connection. The coefficients of this connection with respect to a system of local coordinates are called Christoffel symbols.

Curvature

Recall \(L_[X,Y] = [L_X, L_Y]\). If \([X,Y]=0\) commutes, then \(L_X L_Y - L_Y L_X = 0\) This fails in general for the covariant derivative. This failure is measured by a tensor field that plays a central role in all of differential geometry.

\[R(X,Y) Z = \delta_X \delta_Y Z - \delta_Y \delta_X Z - \delta_[X,Y] Z\]

This is a (1, 3) tensor field which can be proven by showing that \(R\) is \(C^\infty\)-linear with respect to each component.

Lie Groups

See notebook on Lie Groups

https://github.com/lukepereira/notebooks/

E. Meinrenken and G. Gross, Introduction to Differential Geometry, Lecture Notes for MAT367.

Samelson, H. Review: Werner Greub, Stephen Halperin and Ray Vanstone, Connections, curvature, and cohomology. Bull. Amer. Math. Soc. 83 (1977), no. 5, 1011–1015. https://projecteuclid.org/euclid.bams/1183539466

Paul Seidel. 18.950 Differential Geometry. Fall 2008. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

Sigurdur Helgason. 18.755 Introduction to Lie Groups. Fall 2004. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.