Tutorial 1—Global Analysis 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). 1