Homework 2 – Math 549 – Fall 2010
(1) Let M⊂RNand M′⊂RN′be submanifolds. Show that M×M′⊂RN×RN′is a
submanifold.
(2) Show that the special orthogonal group
SO(n) := A∈Mn(R) : det(A) = 1 and ATA= 1
is an n(n−1)
2-dimensional smooth submanifold of Mn(R), where Mn(R) is the vector space
of all n×n-matrices, identified with Rn2.
Hint: Let U⊂Mn(R) be the set of matrices Awith det(A)>0, it is an open set. Define a
map f:U→Sym(n) by f(A) := ATA−1, where Sym(n)⊂Mn(R) consists of all symmetric
matrices. Note that Sym(n) can be identified with R
n(n+1)
2.
(3) Let Mbe a connected topological manifold. Show that Mis path connected.
(4) Recall that the Klein bottle K2is the quotient space of the unit square in [0,1] ×[0,1] with
the identifications indicated in the figure below. Find a group Gacting freely and properly
discontinuously on R2such that K2is homeomorphic to the quotient R2/G by the action
of Gon R2.
(5) Consider the equivalence relation ∼on the complex vector space Cn+1 \{0}given by
z∼w:⇔z=λw for some λ∈C\{0}.
The quotient space CPn:= (Cn+1\{0})/∼, endowed with the quotient topology, is called
complex projective space. The equivalence class of an element (z1,...,zn+1)∈Cn+1\{0}is
denoted by [z1:z2:···:zn+1 ].
(a) Show that CPnis a (2n)-dimensional C∞-smooth manifold and that it can be given a
differentiable structure in such a way that the canonical pro jection
π:Cn+1\{0} → CPn,(z1,...,zn+1)7→ [z1:z2:··· :zn+1 ]
is C∞-smooth.
(b) Show that CP1is diffeomorphic to S2.
(6) Let Pbe a non-constant complex polynomial. Define a map ˜
P:CP1→CP1by ˜
P([z: 1]) :=
[P(z) : 1] and ˜
P([1 : 0]) := [1 : 0]. Show that ˜
Pis differentiable.
(7) Let π:Cn+1\{0} → CPnbe the canonical pro jection from problem (5). The Hopf map
H:S2n+1 →CPn
is the restriction of πto the sphere S2n+1 := {(z1,...,zn+1) : |z1|2+···+|zn+1|2= 1} ⊂
Cn+1\{0}. Let n= 1. Find explicitly a diffeomorphism ϕ:CP1→S2such that ϕ◦His
the map
(u, v)∈S37→ (2 Re(u¯v),2 Im(u¯v),|u|2− |v|2)∈S2,
where ¯vis the complex conjugate.
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