Exercises in Differential Geometry
Universit¨at Regensburg, Winter Term 2015/16
Prof. Dr. Bernd Ammann / Dipl.-Math. Manuel Streil Exercise Sheet no. 0 1. Problem
Let (X, d) be a metric space. Show that a compactly supported continuous func- tion f :X →R is uniformly continuous.
Partitions of Unity
Let M be a smooth manifold. Recall that to any open covering {Vβ}β∈B of M there exists a locally finite refinement {Uα}α∈A together with an subordinated partition of unity {ηα}α∈A. In other words
• {Uα}α∈Ais an open covering ofM,and for anyα∈A we find aβ ∈B such that Uα ⊂Vβ.
• For any p ∈ M there exists a neighbourhood W of p such that there are only finitely many α∈A with W ∩Uα 6=∅.
• supp(ηα)⊂⊂Uα, ηα ≥0 andP
α∈Aηα = 1.Note thatP
α∈Aηα(p) is in fact a finite sum.
2. Problem
LetM be a smooth manifold andA, B ⊂M closed and disjoint subsets.
a) Show that there exists a smooth function f :M → R such that f(A) = 1 and f(B) = 0.
b) Conclude that there are disjoint open subsets U, V ⊂M separating A and B, i.e. A⊂U, B ⊂V and U∩V =∅.
Recall the following:
Let K be a subset of a smooth manifold M and f a map of some subset of M containing K to Rm. We say that f is smooth on K if its restriction to K is locally a restriction of a smooth map, i.e. for every point p ∈ K there exists an open neighbourhood U of p in M and a smooth map F : U → Rm that agrees with f onU ∩K.
3. Problem
Let K be closed subset of a smooth manifold M and f : K → R a smooth function. Thenf is a restriction of a smooth function on M.
4. Problem
LetM be a smooth manifold. Show the following approximation theorem.
a) Let f : M → Rm be a continuous map, smooth on a closed subset K of M, and let ε > 0. Then there exists a smooth map g : M → Rm with
f|K = g|K and kf(p)−g(p)k< ε for all p∈M.
b) Now suppose that M is connected. Use a) two prove that any two points of M can be joined by a smooth curve.
