Smooth Manifolds - Applications

Partitions of unity are one of the most powerful technical tools in differential geometry, allowing us to patch together local data into global objects. They rely essentially on the paracompactness of manifolds.

TheoremExistence of Partitions of Unity

Let $M$ be a smooth manifold and $\{U_\alpha\}_{\alpha \in A}$ an open cover of $M$ . Then there exists a collection $\{\rho_\alpha\}_{\alpha \in A}$ of smooth functions $\rho_\alpha: M \to [0,1]$ such that:

$\text{supp}(\rho_\alpha) \subset U_\alpha$ for each $\alpha$
The collection $\{\text{supp}(\rho_\alpha)\}$ is locally finite
$\sum_{\alpha \in A} \rho_\alpha(p) = 1$ for all $p \in M$

Such a collection is called a partition of unity subordinate to $\{U_\alpha\}$ .

This theorem is fundamental for globalizing local constructions. For instance, it allows us to construct Riemannian metrics by patching together local inner products, or to show that every manifold can be embedded in some Euclidean space.

ExampleBump Functions

The standard bump function on $\mathbb{R}$ is

\varphi(t) = \begin{cases} e^{-1/t} & t > 0 \\ 0 & t \leq 0 \end{cases}

This is smooth everywhere (including at $t=0$ ) and provides the building block for partitions of unity. Modified versions give functions that are 1 on a compact set and 0 outside a slightly larger set.

TheoremWhitney Embedding Theorem

Every smooth $n$ -manifold can be embedded as a closed submanifold of $\mathbb{R}^{2n+1}$ . Moreover, it can be immersed in $\mathbb{R}^{2n}$ .

This remarkable result shows that the abstract definition of manifolds via charts is equivalent to thinking of manifolds as subsets of Euclidean space. It validates our geometric intuition while showing that the intrinsic viewpoint is more powerful.

Remark

The embedding dimension $2n+1$ is sharp for general manifolds. For instance, real projective space $\mathbb{RP}^n$ cannot be embedded in $\mathbb{R}^{2n}$ . However, many specific manifolds embed in much lower dimensions - for example, $S^n \subset \mathbb{R}^{n+1}$ .

TheoremSard's Theorem

Let $f: M \to N$ be a smooth map between manifolds. Then the set of critical values of $f$ (images of points where $df_p$ is not surjective) has measure zero in $N$ .

Sard's theorem is indispensable for transversality arguments. It guarantees that "almost all" points in the target are regular values, which by the implicit function theorem means their preimages are smooth submanifolds.

ExampleApplication to Morse Theory

If $f: M \to \mathbb{R}$ is a smooth function, Sard's theorem implies that almost every real number $c$ is a regular value. Thus the level sets $f^{-1}(c)$ are smooth hypersurfaces for almost all $c$ . Only at critical values can the topology of level sets change.

DefinitionTransversality

Two submanifolds $P, Q \subset M$ are transverse at $p \in P \cap Q$ if

T_pP + T_pQ = T_pM

They are transverse if they are transverse at every point of intersection.

Transverse intersections are generic and well-behaved: if $P$ and $Q$ are transverse, then $P \cap Q$ is itself a smooth submanifold with $\dim(P \cap Q) = \dim P + \dim Q - \dim M$ .