Tutorial 1—Global Analysis
21.09.2021
1. Consider the cylinder in R3
given by the equation
M := {(x, y, z) ∈ R3
: x2
+ y2
= R2
},
where R > 0. Show that M is a 2-dimensional submanifold in R3
. Moreover, give
formula for local parametrizations and local trivializations, and a description of M
as a local graph.
2. Consider a double cone given by rotating a line through 0 of slope α around the
z-axis in R3
. It is given by the equation
z2
= (tan α)2
(x2
+ y2
).
At which points is the double cone a smooth submanifold of R3
? Around the points
where it is give a formula for local parametrizations and trivializations, and a description
of it as a local graph.
3. Denote by Hom(Rn
, Rm
) the nm-dimensional vector space of linear maps from Rn
to Rm
. Consider the subset Homr(Rn
, Rm
) of linear maps in Hom(Rn
, Rm
) of rank
r. Show that Homr(Rn
, Rm
) is a submanifold of dimension of r(n + m − r) in
Hom(Rn
, Rm
).
Hint: Let T0 ∈ Homr(Rn
, Rm
) be a linear map of rank r and decompose Rn
and
Rm
as follows
Rn
= E ⊕ E⊥
and Rm
= F ⊕ F⊥
, (0.1)
where F equals the image of T0 and E⊥
the kernel of T0, and (·)⊥
denotes the
orthogonal complement. Note that dim E = dim F = r. With respect to (0.1) any
T ∈ Hom(Rn
, Rm
) can be viewed as a matrix
T =
A B
C D
,
where A ∈ Hom(E, F), B ∈ Hom(E⊥
, F), C ∈ Hom(E, F⊥
) and D ∈ Hom(E⊥
, F⊥
).
Show that the set of matrices T with A invertible defines an open neighbourhood
of T0 and characterize the elements in this neighbourhood that have rank r (equivalently,
the ones that have an (n − r)-dimensional kernel).
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