October 22, 2019
Having defined smooth manifolds, smooth functions on them and smooth maps between them, we would like to define derivatives. Throughout the note, N and M are smooth manifolds with dimensions n and m, respectively.
1 Tangent Spaces - Abstract Approach
In calculus, we have learned the concepts of tangent spaces at a point p of a space N , with N being various dimensions (line, surface, ...etc.) In all these cases, tangent space to a space N at point p is the best approximation to N by vector space of the same dimension, whose 0 is identified with p. We will therefore define tangent space to a manifold N at p to be some n-dimensional vector space that best approximates. For trivial example, let N = V be an n- dimensional vector space. The tangent space to V at p is simply the translation of V by p. Precisely, it is
TpV = {(p, v) | v ∈ V },
with vector space operations defined by a(p, v) = (p, av) and (p, v1) + (p, v2) = (p, v1+ v2). The multiplication and addition on the right are those for V . We have worked on these spaces for V = Rn when computing DF in advanced calculus, with TpV = Rnp, where
Rnp := {(p, v) | v ∈ Rn}
is the vector space with linear structures defined above. We abbreviate the vector (p, v) by vp. They are referred to as the geometric tangent vectors at p.
Nonlinearly, in elementary single variable calculus, we defined tangent line lp to the graph of a function F : R → R to be the line going through (p, F (p)) with slope F0(p). In another words, it is the 1-dimensional vector subspace of R2(p,F (p)) spanned by vp = (1, F0(p)) whose origin is identified with (p, F (p)) ∈ R2. That is naturally the tangent space for the curve y = F (x). Similar situations arise when we have F : Rn−1 → R, where the tangent space to the graph is the n − 1 dimensional vector subspace of Rn spanned by
Nevertheless, one notices that the vector spaces above are all defined on manifolds with a global chart: Rn, vector space V , graph of a functions. All the ”tangent vectors” are globally identified with ones on Rn by translating to origin and are all explicitly written accordingly. These descriptions become coordinate dependent when identifications with Rnare only defined locally. We wonder naturally whether different coordinate really give ”the same thing”.
Or, more fundamentally, what does the phrase ”vectors tangent to the space”
really mean? As such, we seek alternative definition of tangent vectors that are reasonably coordinate independent and preserve the characteristic property of tangent vectors in Rn, which are then generalized to abstract manifolds.
Take the graph of dimension 1 above for exposition. Every point P = (p, F (p)) is globally identified with p ∈ R. The tangent line is spanned by vector vp= (1, F0(p)). Take a smooth function f on the curve with coordinate representation ˜f . Explicitly, we have
f (x, F (x)) = ˜f (x).
Now take a curve on the graph, which must be of the form γvp(t) = (γ1(t), F (γ1(t))),
with the property that γvp(0) = P and γ0vp(0) = vpor equivalently γ(0) = p and γ0(0) = 1. Basic calculus shows that
d
dt|t=0f (γvp(t)) = ˜f0(p).
Denote the quantity above by vp(f ). It is not hard to verify that for all smooth functions f, g and c ∈ R,
• vp(f + cg) = vp(f ) + cvp(g).
• vp(f g) = f (p)vp(g) + g(p)vp(f ).
Therefore, vp can be viewed as a linear map on the space of smooth functions satisfying some product rule (a.k.a. the Leibiniz rule). We call such a map a derivation at p. One may check that
• The definition vp: C∞→ R does not involve coordinates.
• Constant functions have zero derivation.
More generally, one can show that for every a ∈ R, d
dt|t=0f (γavp(t)) = a ˜f0(p),
where γavp(t) = (γa(t), F (γa(t))) with γa0(0) = a. With this, the space of derivations is itself a vector space.
This view of tangent vectors, as actions on smooth functions, continue to make sense for manifolds. We then define, abstractly and coordinate invariantly, Definition 1.1. Given a smooth manifold N , its tangent space at p denoted by TpN , is the vector space of derivations at p.
The map sending p to Xp ∈ TpN is called a derivation on N , which in elementary cases is called vector field.
2 Tangent Spaces - Concrete Constructions
Let’s take N = Rn and show the expected result that TpN ' Rnp ' Rn for p ∈ Rn.
Proposition 2.1. TpRn' Rnp ' Rn.
We will denote Diby ∂x∂i|p, or ∂pi. They are natural basis for the tangent space of Rn at p.
One notes that the description above is entirely local. We only need to know the objects involved on some neighborhood of p. Therefore, it entirely makes sense to define a tangent space to a manifold at a point, where local coordinate is given by coordinate chart. Next, we construct basis to tangent spaces of a manifold that are represented by local coordinates.
3 The Differentials
Now that we know what tangent spaces are, we are ready to discuss what
”DF ” is for a smooth math F : N → M . Recall that in advanced calculus, the derivative of map at a p is defined to be a linear map between tangent spaces at p and F (p) that best approximates the map near them. We will follow this principle to the general case.
Let’s start with an abstract definition that seems unrelated. Given a smooth map F : N → M and p ∈ N , we define the differential of F at p to be the linear map dF : TpN → TF (p)M by
basic properties are easy to verify and coincide with usual derivatives. For examples,
• d(G ◦ F ) = dG ◦ dF.
• d(IdM) = IdTpM.
• If F is a diffeomorphism, then dF is a vector space isomorphism.
Differentials are indeed generalizations of ordinary derivatives. Let’s dig into the details.
4 Computations in Coordinates
Let’s combine all these abstract constructions and express them in terms we are familiar with. Let (U, φ) be a coordinate chart of p ∈ N so that
ϕ : U → ϕ(U ) := ˜U
is a diffeomorphism (with coordinate representation the identity). Previous results combine together to give isomorphisms between vector spaces below:
TpN ' TpU ' TpU ' T˜ pRn' Rnp ' Rn, where the second isomorphism is dϕ. We denote
∂
∂xi|p:= (dϕ)−1 ∂
∂xi|ϕ(p). More precisely, for smooth function f ∈ C∞(N ),
Clearly, {∂x∂i|p}ni=1 forms a basis for TpN .
It is then a routine exercise to write out the matrix representation of the linear map dFp: TpN → TF (p)M for a smooth map F : N → M :
Just like in linear algebra, different choices of basis amounts to basis changing isomorphisms:
5 The Tangent Bundle
Let’s us finally study something more than ”locally advanced calculus/linear algebra” by collecting all the tangent spaces on a manifold M and form a new space. This is an example of ”vector bundle”, where we attach a vector space to each point of a manifold, all of the same degree, say k. Very often they form new manifolds, and are standard space to describe configurations spaces in physics (especially in electromagnetism).
As a set, the tangent bundle is simply T M = G
p∈M
TpM.
Of course, as sets, T M is always bijective to M × Rm. In fact, it has a natural smooth structure making the projection map smooth.
Theorem 5.1. For any m-dimensional manifold M , its tangent bundle T M is a smooth manifold of dimension 2m so that the projection map π is smooth.
It is important to know whether T M is anything ”new” or just M × Rm as manifolds. The latter case is called a trivial bundle. For M with a global coordinate chart, we observe that T M is always trivial as ˜ϕ in the proof above gives the required diffeomorphism. This is not true for general manifolds M , but are always true locally. That is, π−1(U ) ⊂ T M is always diffeomorphic to U × Rm via the local coordinate chart. Therefore, just like manifolds, which consist of gluing together Euclidean spaces, tangent bundles (in fact vector bundles in general) consist of gluing together trivial bundles.
