Chapter 3 - Tangent Vectors Flashcards
Define: geometric tangent space to R^n at a, geometric tangent vector
How to think about? Vector space?
Given a point a in R^n, the GEOMETRIC TANGENT SPACE to R^n at a, denoted R^n_a is the set {a} x R^n. A GEOMETRIC TANGENT VECTOR in R^n is an element of R^n_a for some a in R^n.
Abbreviate (a, v) as v_a and think of v_a as the vector v with its initial point at a.
Real vector space isomorphic to R^n - just forget about {a} coordinate and do everything as usual in second component. e_i|a are a basis
Discuss point derivations w.r.t. R^n and relation to geometric tangent space
We need a way to generalize the idea of geometric tangent vectors to manifolds that are not embedded in some Euclidean space.
At this point, all we have to work with on smooth manifolds are: smooth maps and smooth coordinate charts.
Recall the common interpretation of tangent vectors as “infinitesimal change”. They provide magnitude and direction for taking directional derivatives. The idea is to characterize tangent vectors by the fact that they can take directional derivatives.
Define a DERIVATION at a to be a linear map w:C^inf(R^n) –> R satisfying the product rule.
Let T_a R^n denote the set of point derivations at a. Show that this is a vector space naturally isomorphic to the geometric tangent space R^n_a. Surjective - use Taylor theorem
As a corollary, the derivations d/dxi form a basis for T_a R^n
Let M be a smooth manifold with or without boundary. Define: tangent space to M at p, tangent vector at p
How to think about?
A linear map v : C^inf(M) –> R is a called a DERIVATION at p if it satisfies product rule. The set of all derivations of C^inf(M) at p, denoted T_pM is a vector space called the TANGENT SPACE to M at p. An element of T_pM is called a TANGENT VECTOR at p.
Should still picture tangent vectors to M as “arrows” that are tangent to M and whose base points are attached to M. Must use abstract def in proofs, but should reason with calc 3 intuition.
Define: differential of F at p
why does this make sense?
Properties?
If M and N are smooth manifold with or without boundary and F:M –> N is a smooth map, for each p in M we define a map dF_p : T_pM –> T_F(p)N called the DIFFERENTIAL of F at p as follows.
Given v in T_pM, we let dF_p(v) be the derivation at F(p) that acts on f in C^inf(N) by the rule:
dF_p(v)(f) = v(f o F)
Show this is in fact a derivation.
Properties of Differentials
- dF_p is linear
- Covariant functor (Manifolds & Smooth maps –> f.d. Vector spaces and Linear maps)
- By (2) diffeomorphism yields isomorphism of tangent spaces
pg 55-56
Show tangent vectors act locally. What does this have to do with tangent spaces?
Let M be a smooth manifold with or without boundary, p in M, and v in T_pM. If f, g in C^inf(M) agree on some neighborhood of p, then vf = vg.
Pf. Let h = f-g a smooth function vanishing in a neighborhood of p. Let psi be a smooth bump function identically 1 on supp h and supported on M \ {p}. Then psih = h and since h(p) = psi(p) = 0, vh = v(psih) = 0. By linearity vf - vg.
This proposition allows us to identify the tangent space to an open submanifold with the tangent space of the whole manifold. i.e. the tangent space is determined by looking in an arbitrarily small neighborhood.
pg 56
Prove: If M is a smooth manifold with or without boundary, U < M is open, and i: U –> M is inclusion, then for every p in U, the differential di_p : T_pU –> T_pM is an isomorphism
Proof. Injectivity: suppose di_p(v) = 0. For any f in C^inf(U), we can extend to a f’ in C^inf(M). Since v acts locally, we have vf = v(f’|U) = v(f’ o i) = di(v)_pf’ = 0. So v = 0.
Surjectivity: similar
So we can identify T_pU with T_pM. di_p(v) is the same derivation as v, thought of as acting on functions on the bigger manifold M instead of functions on U. Since the action of a derivation on a function depends only on the values of the function in an arbitrarily small neighborhood, this is a harmless identification.
pg 56
Prove: If dim M is n, then dim T_pM is n.
What must change for manifold with boundary?
Choose a smooth coordinate chart (U, phi) around p. Since phi is a diffeomorphism, dphi_p is an iso from T_pU to T_phi(p)U hat. Since T_pU = T_pM and T_PU hat = T_pR^n (tangent space depends only on arbitrarily small neighborhoods), it follows that dim T_pM is n.
For a manifold with boundary, we can get an iso from T_pM to T_pH^n using exact same proof as above. Then finish off with the additional fact that the inclusion of H^n –> R^n yields an isomorphism di_p of T_pH^n with T_pR^n.
pg 57-58
Discuss the coordinate vectors at p in M. How do they act? Coordinate basis for T_pM? How can you find the coordinates foer v in T_pM w.r.t. some coordinate basis?
Do an example.
Recall that any coordinate chart (U, phi) yields a diffeomorphism of U with an open subset of R^n. By functorality and locality of differential, this yields an isomorphism T_pM = T_p’R^n.
The standard directional derivatives are a basis for T_p’R^n. We can pull these back to a basis for T_pM using d(phi)_p^-1, d/dxi|p. The vectors d/dxi are the COORDINATE VECTORS at p.
d/dxi|p acts on f in C^inf(M) by taking the ith partial derivative (w.r.t coordinate representations).
Call (d/dxi|p) a COORDINATE BASIS for T_pM. As we have a basis, any v in T_pM can be written uniquely as a linear combo v = v^i d/dxi|p. The numbers v^i are called the COMPONENTS of v w.r.t. the coordinate basis.
If v is known, we can easily compute its components by looking at what it does to the coordinate functions, v(x^j) = v^j.
pg 60-61
Discuss what differentials look like in coordinates. Do an example
First consider case F: U –> V where U < R^n and V < R^m. Chain rule lets us see action - just Jacobian.
Next consider general case of smooth map between smooth manifolds with or without boundary.
pg 61-63
Discuss change of coordinates. Do an example.
Very similar to the differential in coordinate. Essentially just looking at differential of transition map. Comes down to Jacobian again
Look at polar to standard coordinates…
63-65
Discuss the Tangent Bundle and its natural smooth structure. Natural coordinates?
Given a smooth manifold M with or without boundary, we define the TANGENT BUNDLE of M, denoted TM, to be the disjoint union of the tangent spaces at all points of M:
TM = U T_pM
Write an element as ordered pair (p, v) with p in M and v in T_pM.
The tangent bundle comes equipped with a NATURAL PROJECTION map pi: TM –> M which sends each vector in T_pM to the point p at which it is tangent: pi(p,v) = p.
We can define a smooth structure on TM making pi smooth. The charts of this smooth structure…
pg 65 -67
Discuss global differential and its properties
By putting together the differentials of F at all point of M, we obtain a globally define map between tangent bundles called the GLOBAL DIFFERENTIAL and denoted dF : TM –> TN. This is just the map whose restricition to each tangent space T_pM is dF_p.
With the natural smooth structures on TM and TN, DF is a smooth map between tangent bundles.
We have a functor from (smooth manifolds & smooth maps) to (smooth tangent bundles & smooth global differentials)
Define velocity of curve and discuss properties.
How does it act?
First, recall a CURVE in M is just a continuous map gamma: J –> M where J < R is an interval.
Given a smooth curve gamma : J –> M and t0 in J, we define the VELOCITY OF GAMMA at t0, denoted gamma’(t0) to be the vector:
gamma’(t) = d(gamma)(d/dt|t0) in T_gamma(t0)M.
Acts on f as the derivation at gamma(t0) obtained by taking the derivative of f along gamma.
In coordinates, essentially the same formula in euclidean space – tangent vector whose components in a coordinate basis are the derivatives of the component functions of gamma.
pg 69
Show that every tangent vector on a manifold is the velocity vector of some curve
This allows us to think of the tangent bundle as the set of all velocity vectors of smooth curves in M. Choose any chart centered at p. Look at curve gamma with coordinate rep the linear function gamma(t) = vt where v is the desired velocity vector…
pg70
How can we use curves to compute the differential?
First, 3.24 says (F o gamma)’(t0) = dF(gamma’(t0)). Reading this from right to left, we see that if we want to find dF(v), we can find a smooth curve gamma with velocity v at t0. Then dF(v) = dF(gamma’(t0)) = (F o gamma)’(t0).
pg 70
