MAT3009 Manifolds and Topology Semester 1 2019/20
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MAT3009/5/Semester 1 19/20
Manifolds and Topology
Question 1
Let X be a set, and let 7 C p(X) a collection of subsets of X .
(a) (i) State the conditions that 7 must satisfy in order to define a topology on X.
(ii) State the additional condition required in order that the topological space (X, 7)
be Hausdorff.
(b) Let X := {a, b, c} be a set consisting of three elements.
(i) Give a topology 71 on X that is Hausdorff.
(ii) Give a topology 72 on X that is not Hausdorff. State why 72 is not Hausdorff.
(c) Let (X, 7) be a Hausdorff topological space, and (zn )n∈N a sequence of elements of X .
(i) Give the definition of the statement that the sequence (zn )n∈N converges to a point g e X .
(ii) Show that if a sequence (zn )n∈N converges to g e X and also converges to r e X then g = r.
(d) Let N be a set containing n elements, where n > 3. Let 7 be a topology on N with the property that all subsets of N containing n - 1 elements are open.
(i) Show that all subsets of N containing n - 2 elements are open.
(ii) Determine all open subsets of N .
Question 2
(a) Let M be a set.
(i) What does it mean to say that “(U, u) is a chart on M”?
(ii) State the properties that charts (U, u), (V, o) on M must have in order to be com- patible.
(iii) State the properties that a family of charts {(Uα , uα ) I a e A} must have in order to define an atlas on M.
(b) Let M be the ellipse
M := ,(z, y) e 皿2 │ ╱ 、 2 + ╱ 、 2 = 1 、,
where a, b > 0 are real numbers. Let
N := (0, b), s := (0, -b).
denote the North and South poles, and define the open sets
UN := M \ {N}, US := M \ {s}.
Let uN and uS denote the stereographic projection maps,
uN : UN → 皿, uS : US → 皿,
from the North and South poles, respectively.
(i) Determine explicit expressions for uN and uS in terms of the components of points in UN and US respectively.
(ii) For a point g e UN n US , compute uN (g) . uS (g), simplifying the result as much as possible.
(iii) Show that uN o uS(_)1 : uS (UN n US ) → uN (UN n US ) is a smooth, bijective map.
(iv) Show that M is a smooth manifold of dimension 1.
[You may assume, without proof, that the maps uN and uS are bijections.]
(c) Let M, N be smooth manifolds, and f : M → N a map.
(i) Define what it means for the map f : M → N to be smooth (or C& ).
(ii) Let N = 皿, with chart (V, o) where V = 皿 and o : V → 皿 the identity map o(z) = z for all z e V = 皿. Let M be as in part (b), with charts (UN , uN ) and (US , uS ). Show that the map
f : M → 皿; f ((z, y)) := y
is smooth.
Question 3
(a) Let M be a manifold, and g e M .
(i) Let c1 , c2 : 皿 → M be smooth curves with c1 (0) = c2 (0) = g. Define what it means to say that c1 and c2 are tangent (or tangential) at g with respect to a chart (U, u) around g.
(ii) Let (U, u), (V, o) be (compatible) charts around g, and c1 , c2 : 皿 → M smooth curves that are tangential at g with respect to (U, u). Show that c1 and c2 are tangential at g with respect to (V, o).
(iii) Give the definition of a tangent vector to M and g. Define the tangent space to M at g, Tp M.
(iv) Given smooth manifolds M , N and a smooth map f : M → N, give the definition of the tangent map Tp f : Tp M → Tf (p)N.
(b) Let M = 皿3 , with chart (U, u) given by U = 皿3 with u : 皿3 → 皿3 the identity map and Cartesian coordinates (z, y, 3). We identify tangent vectors and vector fields with differential operators o(z, y, 3) + u(z, y, 3) + w(z, y, 3) in the standard way. Let (a, b, c) e 皿3 be fixed, and f : M → M the map
(z, y, 3) 1→ f(z, y, 3) := (a + z, b + a3 + y, c + 3) .
(i) Show that, for all x0 = (z0 , y0 , 30 ) e M, the tangent map T&L f : T&L M → Tf (&L ) M takes the form
(T&L f) ╱ 、 = , (T&L f) ╱ 、 = , (T&L f) ╱ 、 = + a .
(Please state, without proof, and results that you use concerning the coordinate rep-
resentation of the tangent map.)
(ii) Show that the vector fields
e1 :=
property
(T&L f) (ei ) = ei , i = 1, 2, 3.
e 皿3 .
(z, y, 3) e 皿3
(iii) Find the basis {91 , 92 , 93 } for T&(*)L M dual to the basis {e1 , e2 , e3 } for T&L M given
in part (ii).
(iv) Calculate explicitly the (1, 1) tensor field much as possible.
i(3)=1 9i 8 ei , simplifying your result as
Question 4
(a) Let M be a manifold, w e Ωk (M) and 1 e Ωl (M).
(i) State, without proof, the relationship between w A 1 and 1 Aw.
(ii) Let M = 皿n , with coordinates (z1 , z2 , . . . , zn ). Let w e Ωk (M) be given by
w := ← wi6 i| ... ià (z) dzi6 A dzi| A . . . A dzià .
1冬i6 <i|<...<ià冬n
Give the definition of dw .
(iii) Without proof, state the expression for d(w A 1) in terms of dw and d1 .
(b) Let M = 皿3 with Cartesian coordinates (z, y, 3), and
a := A dy A d3 + B d3 A dz + C dz A dy e Ω2 (皿3 )
where A, B, C are smooth functions of (z, y, 3). Find da, writing your result in terms
of the divergence of a vector field on 皿3 .
(c) Let M be a four-dimensional manifold with local coordinates (u, o, n, r). Let
91 = - sin o dn + cos o sin n dr,
92 = cos o dn + sin o sin n dr,
9 = do + cos n dr.
Let 8, 7 e Ω2 (M) be given by
8 := f(u) ╱du A 91 + u 92 A 93、,
7 := g(u) ╱du A 91 - u 92 A 93、,
where the functions f(u), g(u) depend only on the coordinate u .
(i) Compute d91 , d92 , d93 , expressing the results in terms of the forms 91 A 92 , 92 A 93 and 93 A 91 .
(ii) Calculate 8 A 7, simplifying your result as much as possible.
(iii) Determine explicitly all functions f(u) and g(u) for which the two-forms 8 , 7 are closed, i.e. d8 = 0 and d7 = 0.
2022-01-20