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
Discuss the relationship between Lie groups and Lie algebras
Categories
Lie algebras provide a linear model for Lie groups. Similar to how tangent spaces provide linear model for manifolds. Lie groups have more structure than manifolds, so Lie algebras have more structure than tangent spaces.
We want to understand to what extent the structure of a Lie group is preserved by passing to Lie algebra. We will show that there is a functor from the category Lie of Lie groups and Lie group homomorphisms to the category lie of Lie algebras and Lie algebra homomorphisms.
We have already seen that we can associate a Lie algebra to every Lie group - have objects handled. Need to deal with morphisms.
If F : G –> H is a Lie group homomorphism, we need to construct a corresponding Lie algebra homomorphism F_: g –> h. It turns out for every X in g there is a unique vector field in h that is F-related to X. Setting F_X to this vector field, F_* is a Lie algebra homomorphism – called the INDUCED LIE ALGEBRA HOMOMORPHISM.
PROPERTIES
- (Id_G)_* induced by identity map is identity of Lie(G)
- (F2 o F1)* = F2* o F1_1 commutes with compositions
So we have a functor. As usual, this implies that isomorphic Lie groups have isomorphic Lie algebras.
pg 195-196
How can we view Lie(H) as a subalgebra of Lie(G)? Proof?
Notice, elements of Lie(H) are vector fields on H not G so strictly speaking are not elements of Lie(G). What we do have is a Lie subalgebra h < Lie(G) that is canonically isomorphic to Lie(H).
Let i : H –> G be inclusion, then:
h = i_*(Lie(H))
= { X in Lie(G) : Xe in TeH }
Pf. Just follows from properties…pg 197
Show how to compute the Lie algebra of O(n)
pg 197
What is Ado’ Theorem? Corollaries?
Recall: A f.d. representation of a Lie algebra is a homomorphism phi : g –> gl(V)
Thm. Every finite dimensional realm Lie algebra admits a faithful f.d. representation
Cor. Every f.d. real Lie algebra is isomorphic to a Lie sub algebra of some matrix algebra gl(n,R) with the commutator bracket.
Recall, the analogous result for Lie groups is false. There are Lie groups, that are not isomorphic to Lie subgroups of GL(n,R).
