Chapter 8 - Vector Fields Flashcards
Define: vector field on M, smooth, rough, support, component functions of X
If M is a smooth manifold, a VECTOR FIELD ON M is a section of the map pi : TM –> M, i.e. a continuous map X : M –> TM usually written p -> X_p with the property that pi o X = Id_M, ie X_p in TpM for each p in M.
visualize as an arrow attached to each point of M, chosen to be tangent to M and to vary continuously from point to point
SMOOTH VF if smooth as map from M to TM, ROUGH VF not continuous.
SUPPORT OF X is the closure of {p in M : X_p != 0}
If X : M –> TM is a rough vector field and (U, (x^i)) is any smooth coordinate chart for M, we can write the value of X at any point p in U in terms of the coordinate basis vectors:
X_p = X^i(p) d/dx^i |p
This defines n functions X^i : U –> R called the COMPONENT FUNCTIONS OF X in the given chart.
pg 174-175
How does smoothness of component functions of a vector field relate to smoothness of the vector field?`
Prop. Let X: M –> TM be a rough vector field. If (U, (x^i)) is any smooth coordinate chart on M, then the restriction of X to U is smooth <=> its component functions w.r.t. this chart are smooth
Pf. Let (x^i, v^i) be the natural coordinates on pi^-1(U) < TM associated with the chart (U, (x^i)). Then the coordinate rep of X on U is (x1, … , xn, X^1(x), …, X^n(x)).
pg 175
Examples of vector fields?
- Coordinate vector fields:
Given any smooth chart (U, (x^i)) on M, the assignment p –> d/dx^i |p determines a vector field on U called the ith COORDINATE VECTOR FIELD and denoted by d/dx^i – smooth because coordinate functions are constants - The Euler Vector Field:
The vector field V on R^n whose value at x in R^n is V_x = x^1 d/dx^1|x + … + x^n d/dx^n|x – smooth because coordinate functions are linear - The Angle Coordinate Vector Fields on Circle/Tori:
d/dtheta is a globally defined vector field on S^1 coming from angle coordinates. Any angle coordinates yield same vector field on common domain. Do the same thing on T^n to define smooth global coordinates on T^n
Define: (smooth) vector field along A
Discuss extending vector fields from subsets
If M is a smooth manifold and A < M is an arbitrary subset, a VECTOR FIELD ALONG A is a continuous map X : A –> TM satisfying pi o X = Id_A (ie X_p in TpM for each p in A). Call it SMOOTH if if we can extend it to a smooth vector field on a neighborhood V or p in U.
Lemma. Let A < M be a closed subset and X smooth vf along A. Given any open subset U containing A, there exists a smooth global vector field X Tilda on M s.t. X tilda | A = X and supp X Tilda < U. Prove it.
Compare to Lemma 5.34
Prop. Given p in M and v in TpM, there is a smooth global vector field X on M s.t. Xp = v.
Discuss the set of smooth vector fields on M. Structure?
Denote this xi(M). This is a module over the ring C^inf(M). Notice C^inf(M) is an algebra over R. Actually a Lie algebra…
Define: linearly independent VFs, span tangent bundle, local frame for M, global frame, smooth frame
An ordered k-tuple (X1, … , Xk) of vector fields defined on some subset A < M is s.t.b LINEARLY INDEPENDENT if (X1p, … Xkp) is linearly independent k-tuple in TpM for each p in A, SPANS THEN TANGENT BUNDLE if X1p, … Xkp) spans TpM for all p in A.
A LOCAL FRAME for M is an ordered n-tuple of vector fields (E1, … , En) defined on an open subset U < M that is linearly independent and spans the tangent bundle, ie (E1p, … , Enp) form a basis for TpM. GLOBAL FRAME if U = M. SMOOTH FRAME if each of the vector fields Ei is smooth.
Examples of Local and Global Frames
- The standard coordinate vector fields form a smooth global frame for R^n
- If (U, x^i) is any smooth coordinate chart, the coordinate vector fields form a smooth local frame on U called the COORDINATE FRAME
- The vector fields defined on S^1 and T^n using angle coordinates are smooth global frames
Discuss completion of local frames
Let M be a smooth n-manifold
1. COMPLETE INDEPENDENT VF: If (X1, …, Xk) is a linearly independent k-tuple of smooth vector fields on an open subset U < M, then for each p in U there exist smooth vector fields X_k+1, … , X_n in a neighborhood V of p s.t. (X1, … , Xn) is a smooth local frame on U int V.
- COMPLETE INDEPENDENT TANGENT VECTORS: If (v1, … , vk) is a linearly independent k-tuple of vectors in TpM for some p in M, there exists a smooth local frame (Xi) on a neighborhood of p s.t. Xi|p = vi for I = 1, …,k.
- If (X1, … , Xn) is a linearly independent n-tuple of smooth vector fields along a closed subset A < M, then there exists a smooth local frame on some neighborhood of A that extends Xi, ie X Tilda_i|A = X_i.
Discuss orthonormal frames
For subsets of R^n, we have a natural inner product to work with. Vector fields are orthonormal if assign orthonormal vectors to each point in domain. A frame consisting of orthonormal vector fields is called an ORTHONORMAL FRAME
Using the Gram-Schmidt algorithm, we can turn any local frame into an orthonormal frame
What is a parallelizable manifold? Discuss examples
We always have smooth local frames, but global ones are much rarer. A smooth manifold is PARALLELIZABLE if it admits a smooth global frame.
R^n, S^1, T^n are easily seen to be parallelizable. S^3, S^7 are also parallelizable. All Lie groups are parallelizable.
Most smooth manifolds are not parellelizable.
S^1, S^3, S^7 only parallelizable spheres – only S^1 & S^3 have Lie group structure
Discuss viewing vector field as operator on space of smooth real valued functions. How does this yield alternative characterizations of smoothness for vector fields? Proof?
If X is a vector field on M, and f in C^inf(U), we obtain a new function Xf : U –> R defined by
(Xf)(p) = X_pf
“take derivative at each point in direction of vector field”
TFAE
- X is smooth
- For every f in C^inf(M), the function Xf is smooth on M
- For every open subset U < M and every f in C^inf(U), the function Xf is smooth on U
Pf. (1) => (2) Look at Xf in coordinates (2) => (3) Bump function/tangent vectors act locally (3) => (1) coordinates/apply to coordinate function
pls 180-181
Define: derivation. Relationship with smooth vector fields? Proof?
A DERIVATION is a map X: C^inf(M) –> C^inf(M) that is linear over R and satisfies the product rule X(fg) = fXg + gXf for all f, g in C^inf(M)
Prop. Let M be a smooth manifold. A map D : C^inf(M) –> C^inf(M) is a derivation <=> it is of the form Df = Xf for some smooth vector field X
Pf. First, every smooth vf induces a derivation. Clearly linear and everything follows from properties of tangent vectors at a point.
Conversely, if D is a derivation. Define a vf X by X_pf = (Df)(p).
Thus we can IDENTIFY smooth vector fields on M with derivations of C^inf(M).
pg 181
Discuss using a smooth map F: M –> N to transfer a vf from M to N.
F-related?
If X is a vf on M, then for any p in M, we can obtain a vector dFp(Xp) in T_F(p)N. In general this does not define a vf on N:
- If F not surjective, no way to decide what vector to assign to points outside of codomain
- If F not injective, then for some points of N there may be several different vectors obtained by applying dF to X at different points of M
If there is a vector field Y on N with the property that for each p in M, dFp(Xp) = Y_F(p), the vector fields X and Y are F-related
How do F-related vf act on smooth functions? Pf?
Let X in xi(M), Y in xi(N). Then X and Y are F-related <=> for every smooth real-valued function f defined on an open subset of N,
X(f o F) = (Yf) o F.
Pf. Just unravel both sides using previous definitions pg 182
Discuss vector fields of diffeomorphic manifolds M and N
Pushforward?
If F: M –> N is a diffeomorphism, then for every X in xi(M) there is a unique Y in xi(N) that is F-related to Y. i.e. we have a bijection between sets of vector fields.
Pf. Define Y by Yq = dF_F^-1(q)(X_F^-1(q)). i.e. we look at the vector at the presage of q and sent it to TqN via dF. Clearly unique rough VF F-related to X. Composition of smooth maps
Call Y the PUSHFORWARD OF X BY F and denote F_*X.
When does a VF on a manifold M restrict to a VF on a submanifold S?
If X is VF on manifold M, X does not necessarily restrict to S since X_p may not lie in the subspace TpS < TpM. Given p in S, say X is TANGENT TO S AT P if Xp in TpS and TANGENT TO S if tangent to S at every point of S.
Prop. X is tangent to S <=> (Xf)|S = 0 for every f in C^inf(M) s.t. f|S = 0.
The main prop is the following:
Let S in M be an immersed sub manifold and i: S –> M be inclusion map. Y in xi(M) is tangent to S <=> there is a unique smooth vector field on S that is i-related to Y.
Pf. <= Suppose there is a vf X in xi(S) that is i-related to Y. Then Y is tangent to S because Yp = dip(Xp) is the image of dip for each p in S.
=> Say Y is tangent to S. Then by def, Yp is the image of dip for each p in S. Thus, there is a unique (dip injective) vector Xp in TpS s.t. Yp = dip(Xp). Defines a rough vector field X on S. Just need to show X is smooth. To do this, recall an immersed sub manifold is locally embedded and use slice coordinates… pg 184 - 185
Discuss combining two vector fields
Example?
Lie bracket?
Propeties?
Given a smooth function f: M –> R, we can apply X to f, obtaining another smooth function Xf. Then we can apply Y to this function, obtaining a smooth function YXf. The operation f –> YXf does not generally satisfy the product rule – thus can’t be a vector field.
Example. X = d/dx and Y = xd/dy on R^2. f(x,y) = x, g(x,y) = y. Then XY(fg) = 2x but fXYg + gXYf = x. Doesn’t satisfy product rule
The LIE BRACKET OF X AND Y, {X,Y] is defined by [X,Y]f = XYf - YXf.
Lemma. The Lie bracket of any pair of smooth vector fields is a smooth vector field.
Pf. Show [X,Y] is a derivation of C^inf(M). Just compute…
PROPERTIES
- Bilinearity
- Antisymmetry
- Jacobi Identify
- [fX, gY] = fg[X,Y] + (fXg)Y - (gYf)X
Discuss the coordinate formula for Lie bracket
Do an example.
Let X = X^i d/dx^i and Y = Y^j d/dx^j be the coordinate expressions for X and Y in terms of some smooth local coordinates (x^i) for M. Then [X,Y] has the following coordinate expression:
[X,Y] = (X^i dY^j/dx^i - Y^i dX^j/dx^i) d/dx^j
or more concisely,
[X,Y] = (XY^j - YX^j)d/dx^j
Pf. Compute…
pg 186-187
In what sense is the Lie bracket natural?
Pf?
Corollaries?
Let F : M –> N be smooth, X1, X2 in xi(M), Y1, Y2 in xi(N). If X1 is F-related to Y1 and X2 is F-related to Y2, then [X1, X2] is F-related to [Y1, Y2].
Compute X1X2(f o F) …
Cor. (Pushforward of Lie Brackets) If F: M–> N is a diffeomorphism, then F_[X1, X2] = [F_X1, F_*X2]
Cor. If Y1, Y2 are tangent to S, then [Y1, Y2] is also tangent to S.
Pgs 188-189
Define: left invariant vector field on G
Important properties?
A vector field X on G is said to be LEFT-INVARIANT if it is invariant under all left translations in the sense that it is Lg-related to itself for every g in G.
i.e. d(Lg)g’(Xg’) = Xgg’ for all g, g’ in G
Since Lg is a diffeomorphism, this can be abbreviated by writing (Lg)_*X = X for every g in G.
The set of all smooth left invariant vector fields on G is a LIE ALGEBRA: a real vector space g endowed with a map called the BRACKET from g x g –> g denoted [X, Y] that is:
- Bilinear
- Antisymmetric
- Jacobi Identity
The fact that g is closed under bracket follows from the Naturally of Lie Bracket
Simple examples of Lie algebras? General
- The space xi(M) of all smooth vector fields on a smooth manifold M
- If G is a Lie group, the set of all smooth left-invariant vector fields on G, called the LIE ALGEBRA OF G, denoted Lie(G)
- M(n,R) under COMMUTATOR BRACKET [A,B] = AB - BA. Denote as gl(n,R). Similarly gl(n,C), gl(V)
- ABELIAN Lie algebras. Just set [,] = 0 for all elements of a vector space V
What is the dimension of Lie(G)? Proof?
Thm. The evaluation map epsilon: Lie(G) –> TeG given by epsilon(X) = Xe is a vector space isomorphism. Thus, Lie(G) is finite-dimensional, with dimension equal to dim G.
Pf. Epsilon clearly LINEAR.
INJECTIVE: if epsilon(X) = Xe = 0 for some X in Lie(G), then left-invariance of X implies that Xg = d(Lg)e(Xe) = 0 for every g in G, so X = 0.
SURJECTIVE: Let v in TeG be arbitrary and define a (rough) vector field v^L on G by:
v^L|g = d(Lg)e(v)
This is the only choice for a left-invariant vector field on G with value at the identity = v. It is entirely determined by vector at identity.
v^L Smooth: Just check v^Lf is smooth whenever f in C^inf(G) … have to work a bit for this
v^L Left-invariant: Just compute…
Cor. Every left-invariant rough vector field on a Lie group is smooth.
Define: left-invariant frame
importance for Lie groups?
If G is a Lie group, a local or global frame consisting of left-invariant vector fields is called a LEFT-INVARIANT FRAME.
Prop. Every Lie group admits a left-invariant smooth global frame, and therefore every Lie group is parallelizable.
Pf. If G is a Lie group, every basis for Lie(G) is a left-invariant smooth global frame for G
Show how to determine Lie algebras of: R^n, S^1, GL(n,R)
- R^n: Let Lb(x) = x + b. Then d(Lb) is represented by the identity matrix. Vector field left invariant <=> coefficients are constants. See that Lie(R^n) = R^n
- S^1: Choosing angle coordinates well, each left translation has local coordinate rep of form theta –> theta + c. Again, differential is 1x1 identity matrix, Lie(S^1) = R
- GL(n ,R) : For any Lie group G, we know there is a vector space isomorphism between Lie(G) and the tangent space to G at the identity. So we have Lie(GL(n,R)) = T(GL(n,R))_e. Now since GL(n,R) is an open subset of gl(n,R), its tangent space is naturally isomorphic to the vector space gl(n,R). Thus, composition of isomorphisms gives us Lie(GL(n,R)) = gl(n,R) as vector spaces.
These two vector spaces have independently defined Lie algebra structures - the first coming from Lie bracket of vector field, the second from commutator brackets of matrices. It turns out that the vector space isomorphism above is in fact a Lie algebra isomorphism. pg 193
