Download 3 Topology of a manifold

THEODORE VORONOV
§3
DIFFERENTIABLE MANIFOLDS. Fall 2009
Topology of a manifold
Last updated: November 17, 2009.
3.1
Basic topological properties of manifolds
Consider a manifold 𝑀 = (𝑀, 𝒜), where 𝒜 = (𝜑 : 𝑉 → 𝑈 ) is a smooth atlas
defining the manifold structure on the set 𝑀 . Here 𝑉 ⊂ ℝ𝑛 and 𝑈 ⊂ 𝑀 .
Let us assume that this atlas is maximal, i.e., it is the union of all equivalent
atlases.
Definition 3.1. A subset 𝐴 ⊂ 𝑀 is open if and only if for each chart in the
maximal atlas 𝒜, the set 𝜑−1 (𝐴 ∩ 𝑈 ) ⊂ ℝ𝑛 is open.
Example 3.1. The codomain 𝑈 of each chart is an open set.
Consider unions and intersections of open subsets of a manifold. Let
𝜑 : 𝑉 → 𝑈 be a particular chart. (For the simplicity of notation we have
dropped an index 𝛼.) For an arbitrary family of sets 𝐴𝜇 ⊂ 𝑀 we have
∪
∪
∪
𝜑−1 (( 𝐴𝜇 ) ∩ 𝑈 ) = 𝜑−1 ( (𝐴𝜇 ∩ 𝑈 )) =
𝜑−1 (𝐴𝜇 ∩ 𝑈 )),
∩
∩
∩
𝜑−1 (( 𝐴𝜇 ) ∩ 𝑈 ) = 𝜑−1 ( (𝐴𝜇 ∩ 𝑈 )) =
𝜑−1 (𝐴𝜇 ∩ 𝑈 )).
because 𝜑 is a bijection.
From here follows a theorem:
Theorem 3.1. Open subsets of a manifold 𝑀 (defined as above) satisfy the
axioms of a topology.
Therefore each manifold can be considered as a topological space. We
shall refer to the topology defined above as to the manifold topology or the
topology given by a manifold structure.
Recall the notion of an open cover of a topological space 𝑋. It is a
collection of open sets 𝔘 = (𝑈𝛼 ) such that ∪𝑈𝛼 = 𝑋.
Example 3.2. If 𝒜 = (𝜑𝛼 : 𝑉𝛼 → 𝑈𝛼 ) is an atlas for a manifold 𝑀 , then
(𝑈𝛼 ) is an open cover of 𝑀 , which is often referred to as a ‘coordinate cover’.
There is a useful lemma:
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DIFFERENTIABLE MANIFOLDS. Fall 2009
Lemma 3.1. Suppose (𝑈𝛼 ) is an open cover of 𝑋. A subset 𝐴 ⊂ 𝑋 is open
if and only if 𝐴 ∩ 𝑈𝛼 is open for all 𝛼.
Proof.
∪ If 𝐴 is open, then all intersections with 𝑈𝛼 are open. Conversely,
𝐴 = 𝛼 (𝐴 ∩ 𝑈𝛼 ), which is a union of open sets and thus open.
For manifolds we can say more. Suppose 𝒜′ = (𝑓𝛼 : 𝑉𝛼 ) → 𝑈𝛼 is a
particular atlas for 𝑀 , not necessarily maximal.
Lemma 3.2. A set 𝑂 ⊂ 𝑀 is open if 𝜑−1
𝛼 (𝑂 ∩ 𝑈𝛼 ) are open, for all charts
in 𝒜′ .
Therefore it is sufficient to check the intersections with the codomains for
a particular atlas!
Suppose there is already some topology on the set 𝑀 . Under which
condition will it coincide with the topology induced by a manifold structure
(the “manifold topology”)?
Theorem 3.2. A topology 𝜏 on 𝑀 coincides with the manifold topology if
and only if for some atlas all sets 𝑈𝛼 are open in 𝜏 and all maps 𝜑𝛼 are
homeomorphisms w.r.t. 𝜏 .
Proof. Consider the manifold topology. The sets 𝑈𝛼 are open by definition
of this topology. Why 𝜑𝛼 is continuous ? Take any 𝑉 ⊂ 𝑉𝛼 . 𝜑−1
𝛼 (𝑉 ) ⊂ 𝑈𝛼 .
−1
−1
If 𝑉 = 𝜑𝛼 (𝜑𝛼 (𝑉 )) is open, then by definition 𝜑𝛼 (𝑉 ) is open.
Conversely, let 𝑈𝛼 be open in some 𝜏 and 𝜑𝛼 be homeomorphisms w.r.t.
𝜏 . Let 𝑂 be an open set in 𝜏 . Then 𝑂 ∩ 𝑈𝛼 is open, and since 𝜑𝛼 is a
homeomorphism, then 𝜑𝛼 (𝑂 ∩ 𝑈𝛼 ) is also open. That means, 𝑂 is open in
the manifold topology. Now let 𝑂 be open in the manifold topology. Thus
𝜑𝛼 (𝑂) is open in ℝ𝑛 . Since 𝜑𝑎 are homeomorphisms with respect to 𝜏 , it
follows that all intersections 𝑂 ∩ 𝑈𝛼 are open in 𝜏 . Applying Lemma 3.1, we
obtain that 𝑂 is open in 𝜏 .
Example 3.3. The manifold topology of 𝑆 𝑛 coincides with the usual (subspace) topology.
Consider 𝑆 𝑛 ⊂ ℝ𝑛+1 with the induced topology. Since subsets 𝑆 𝑛 ∖ 𝑁
and 𝑆 𝑛 ∖ 𝑆 are open in this topology, and the formulas for stereographic
projection are given by rational functions (both direct and inverse maps),
then by Theorem 3.2 the subspace topology and the topology induced by
manifold structure coincide.
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Continuity: in terms of neighborhoods, hence in coordinates.
What is the relation between smoothness and continuity?
Theorem 3.3. A smooth map is continuous.
Proof proves follows from the definition of continuity in terms of neighborhoods.
Recall such fundamental notions as compactness and connectedness.
Recall that a topological space is called compact, if one can extract a finite
subcover from its any open cover. Note that a subspace 𝑆 of a topological
space 𝑋 is compact (in induced topology), iff one can extract finite ∪
subcover
of any open cover of 𝑆 in 𝑋, i.e., a collection (𝑈𝛼 ) such that 𝑆 ⊂ (𝑈𝛼 ).
We shall use the following properties of compactness.
Theorem 3.4. Any closed subspace of a compact space is compact.
Proof. Consider an open cover of 𝑆 in 𝑋. Add 𝑋 ∖ 𝑆 to this collection. Then
we have an open cover of 𝑋. Extract a finite subcover and for aesthetical
purposes throw away 𝑋 ∖ 𝑆, if it is still present. This is a finite subcover of
𝑆.
Theorem 3.5. Let 𝑓 : 𝑋 → 𝑌 be continuous, 𝑋 compact. Then 𝑓 (𝑋) is
compact.
Proof. Let (𝑈𝛼 ) be an open cover of 𝑓 (𝑋). Then 𝑓 −1 (𝑈𝛼 ) is an open cover
−1
of ∪
𝑋. Extract a ∪finite subcover 𝑓∪
(𝑈𝛼𝑘 ), 𝑘 = 1, . . . , 𝑁 . Then 𝑓 (𝑋) =
𝑓 ( 𝑓 −1 (𝑈𝛼𝑘 )) = 𝑓 (𝑓 −1 (𝑈𝛼𝑘 )) = 𝑈𝛼𝑘 .
Theorem 3.6 (Heine–Borel). A subspace of ℝ𝑛 is compact, iff it is closed
and bounded (i.e. is contained in some ball of radius 𝑅 < ∞).
Examples.
𝑆 𝑛 is compact.
Now we shall discuss properties of manifolds related to connectedness.
...............................
Example 3.4. 𝑆 𝑛 is connected.
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3.2
DIFFERENTIABLE MANIFOLDS. Fall 2009
Partitions of unity and embedding into ℝ𝑁
We have to admit an embarrassing fact: for a general manifold 𝑀 we do not
have tools allowing to show the existence of smooth functions defined everywhere on 𝑀 (besides constants). On ℝ𝑛 we have plenty of functions: first of
all, the coordinate functions 𝑥𝑖 , then polynomials and various other smooth
functions of 𝑥1 , . . . , 𝑥𝑛 . By contrast, in the absence of global coordinates,
how one can find a non-trivial smooth functions on a manifold?
If, however, a manifold 𝑀 𝑛 is “embedded” into some ℝ𝑁 as some kind of
multidimensional surface (later we shall explain it precisely), then there will
be plenty of smooth functions: all restrictions of the standard coordinates
𝑦 1 , . . . , 𝑦 𝑁 on 𝑀 𝑛 . So the questions about having “enough smooth functions”
and about the possibility to embed a manifold into a Euclidean space are
closely related.
Let us make the following observation. For a manifold (or any topological
space) to be realized as a subspace of ℝ𝑁 , it must satisfy some a priori
topological conditions.
Recall that a topological space is Hausdorff if for any two points 𝑥 ∕= 𝑦
it is possible to find open sets 𝑈𝑥 ∋ 𝑥 and 𝑈𝑦 ∋ 𝑦 s.t. 𝑈𝑥 ∩ 𝑈𝑦 = ∅.
Example 3.5. All metric spaces are Hausdorff, in particular ℝ𝑛 , and any
subspace of a Hausdorff space is Hausdorff. Hence all examples of manifolds
embedded in ℝ𝑁 (“multidimensional surfaces”) are Hausdorff.
Recall that a base of a topological space is a family of open sets such that
an arbitrary open set is the union of sets from that family.
Example 3.6. Suppose 𝑋 is a metric space (e.g. ℝ𝑛 ). The collection of all
open balls is a base.
Example 3.7. For ℝ𝑛 it is possible to show that it is sufficient to take
all open balls with rational centers (the coordinates of the centers must be
rational numbers) and rational radii.
A topological space is called second-countable if it has a countable base 1 .
As we have seen, ℝ𝑛 is second-countable. Clearly, any subspace of a secondcountable space is second-countable too.
1
This traditional terminology comes from certain “first” and “second” “countability
conditions” considered in set-theoretic topology.
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Definition 3.2 (Amended). A manifold from now on is what we have previously called a manifold satisfying, in addition, the requirements that as a
topological space it is Hausdorff and second-countable.
Remark 3.1. A manifold is second-countable if and only if it has a countable atlas. Indeed, suppose it is second-countable. That means there is a
countable base ℬ = (𝐵𝑘 ). Consider an arbitrary atlas (𝜑𝛼 : 𝑉𝛼 → 𝑈𝛼 ). Then
every open set 𝑈𝛼 is the union of some sets 𝐵𝑘 ; hence these particular 𝐵𝑘 ’s
admit local coordinate systems as the restrictions of the coordinate system
on 𝑈𝛼 . Thus we arrive at a new atlas the codomains of which are certain
elements of the family ℬ = (𝐵𝑘 ). In particular, it is countable. Conversely,
suppose there is a countable atlas on a manifold 𝑀 𝑛 . To construct a countable base for 𝑀 𝑛 , take countable bases for the domains of each chart (as an
open subset of ℝ𝑛 ), consider their images in 𝑀 𝑛 and take their union. It is
a countable family of open sets in 𝑀 𝑛 and every open set in 𝑀 is the union
of some elements of this family (since it is the union of its intersections with
the chart codomains).
With these conditions imposed it is possible to show that there is enough
smooth functions. It turns out that it is necessary and sufficient to impose
these topological restrictions on a manifold 𝑀 to guarantee a good supply of
C ∞ functions. It is done below. (Without them, one can construct “pathological” examples where the only smooth functions are constants. On the
other hand, with the restrictions described below, we shall be able to show
that a manifold can be embedded into a Euclidean space of sufficiently large
dimension, which guarantees an abundance of smooth functions.)
Recall that the support of a function (notation: Supp 𝑓 ) is the closure of
the subset where the function does not vanish.
Lemma 3.3 (Bump functions). For any point 𝒙 ∈ 𝑀 of a smooth manifold
𝑀 there is a nonnegative smooth function 𝑔𝒙 ∈ 𝐶 ∞ (𝑀 ) (a “bump function”)
compactly supported in a neighborhood of 𝒙 and which is identically 1 on a
smaller neighborhood.
Proof. First let us prove the following. Let 𝐶𝑎 be a cube {∣𝑥𝑖 ∣ ≤ 𝑎, ∀𝑖} in
ℝ𝑛 . There is a 𝐶 ∞ -function which equals zero outside 𝐶2 and equals one on
𝐶1 (this will be the “local” part of the lemma).
Let 𝑥 ∈ ℝ. Consider the function 𝑓 (𝑥) defined as 𝑒−1/𝑥 for 𝑥 > 0 and zero
for 𝑥 ≤ 0. It is 𝐶 ∞ . Define ℎ(𝑥) := 𝑓 (𝑥)/(𝑓 (𝑥)+𝑓 (1−𝑥)). Since 𝑓 (1−𝑥) = 0
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for 𝑥 > 1, the function ℎ is identically 0 for 𝑥 ≤ 0 and identically 1 for 𝑥 ≥ 1.
Then the function 𝑔(𝑥) := ℎ(𝑥 + 2)ℎ(2 − 𝑥) is zero for 𝑥 ≥ 2 or 𝑥 ≤ −2
and equals 1 for ∣𝑥∣ ≤ 1. Taking the product of such functions for each
coordinate, we obtain the required function on ℝ𝑛 .
Transferring this to a manifold relies on the Hausdorff condition.
The details are as follows. Consider a manifold 𝑀 and a point 𝒙 ∈ 𝑀 . Consider
a coordinate neighborhood 𝑈 around 𝒙. From the local construction we have a smooth
function 𝑔𝒙 defined on 𝑈 such that it is 1 near 𝒙 and its support is contained in some
open 𝑂 ⊂ 𝑈 homeomorphic to an open ball or a cube. In particular, the closure of 𝑂
is homeomorphic to a closed ball, hence is compact. Extend 𝑔𝒙 from 𝑈 to 𝑀 ∖ 𝑈 by
zero. It is necessary to prove that the result is smooth. It is sufficient to prove that for
any 𝒚 ∈ 𝑀 ∖ 𝑈 there is a whole neighborhood 𝑊 on which 𝑔𝒙 is identically zero. Fix 𝒚.
¯ Since 𝑀 is Hausdorff, there are disjoint open neighborhoods
Consider all points 𝒛 ∈ 𝑂.
¯ is compact, we can extract a finite number
𝑂𝑦𝑧 and 𝑂𝑦𝑧 of 𝒛 and 𝒚 respectively. Since 𝑂
¯ Then the intersection ∩ 𝑂𝑦𝑧 is an open
of neighborhoods 𝑂𝑦𝑧𝑘 , 𝑘 = 1, . . . , 𝑁 covering 𝑂.
𝑘
¯ So we can take it as 𝑊 ∋ 𝑦, and the
neighborhood of 𝒚 and does not intersect with 𝑂.
function 𝑔𝒙 is identically zero on 𝑊 , hence smooth at 𝒚.
From the proof it follows that it is possible to construct a bump function
such that it is compactly supported in a given neighborhood of 𝒙.
Having bump functions on a manifold is already sufficient for showing that
there are many global smooth functions. Indeed, take an arbitrary smooth
function defined in a neighborhood of 𝒙 ∈ 𝑀 . Then, by multiplying it by
a suitable bump function (compactly supported inside a smaller neighborhood), it is possible to extend it by zero to the whole manifold 𝑀 . We shall
use this many times in the future.
A collection of subsets 𝐴𝜇 ⊂ 𝑋 is said to be locally finite if each point
𝑥 ∈ 𝑋 has a neighborhood with the empty intersection with all 𝐴𝜇 except
for a finite number of indices 𝜇.
Definition 3.3. A partition of unity on a topological space 𝑋 is a collection
of nonnegative continuous functions (𝑓𝜇 ) such that the collection of their
supports (Supp 𝑓𝜇 ) is locally finite, and for any 𝑥 ∈ 𝑋
∑
𝑓𝜇 (𝑥) = 1.
(1)
𝜇
(For each point the sum is actually finite.)
A partition of unity (𝑓𝜇 ) is called subordinate to a cover (𝑈𝛼 ) if for each
𝜇 there is an open set 𝑈𝛼 , where 𝛼 = 𝛼(𝜇), such that Supp 𝑓𝜇 ⊂ 𝑈𝛼 . We say
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that a partition of unity (𝑓𝛼 ) is subordinate to an open cover (𝑈𝛼 ) with the
same set of indices if Supp 𝑓𝛼 ⊂ 𝑈𝛼 for each 𝛼.
Obviously, everything simplify greatly for finite covers and finite partitions of unity.
For manifolds, we shall look for smooth partitions of unity.
Theorem 3.7. For any open cover (𝑈𝛼 ) of a smooth manifold 𝑀 there exists
a smooth partition of unity subordinate to this cover. It can be chosen with
one of the two additional properties: either
(1) the partition of unity (𝑓𝑘 ) is countable and all functions 𝑓𝑘 belonging
to it are compactly supported;
or
(2) the partition of unity is with the same set of indices, (𝑓𝛼 ) so that
Supp 𝑓𝛼 ⊂ 𝑈𝛼 , for all 𝛼.
In the latter case only a countable number of functions are not identically
zero.
Proof. A proof is based on the existence of bump functions and uses, in
addition, the condition of second-countability. We shall give a proof only for
case when the manifold 𝑀 is compact. It is not necessary for the statement,
but the proof simplifies greatly.
Take an open over of 𝑀 , (𝑈𝛼 ). For each point 𝜉𝛼 ∈ 𝑈𝛼 consider a bump
function 𝑔𝜉𝛼 chosen in such a way that Supp 𝑔𝜉𝛼 ⊂ 𝑈𝛼 . There are open
neighborhoods 𝑊𝜉𝛼 ∋ 𝜉𝛼 in 𝑈𝛼 such that 𝑔𝜉𝛼 = 1 on 𝑊𝜉𝛼 . Consider the
collection (𝑊𝜉𝛼 ) for all 𝛼 and all 𝜉𝛼 ∈ 𝑈𝛼 as an open cover of 𝑀 (it is a
“refinement” of the cover (𝑈𝛼 )). We can extract a finite subcover. Denote
its elements as 𝑊𝑘 , where 𝑘 = 1, . . . , 𝑁 , so that 𝑊𝑘 = 𝑊(𝜉𝛼 )𝑘 ; we have
𝑊𝑘 ⊂ 𝑈𝛼𝑘 . Note that 𝑔𝑘 := 𝑔(𝒙𝛼 )𝑘 is > 0 on 𝑊𝑘 . We define
𝑓𝑘 :=
𝑔𝑘
.
𝑔1 + . . . + 𝑔𝑁
It is well-defined because all 𝑔𝑘 are non-negative
and at each point at least
∑
one 𝑔𝑘 is positive. Clearly, 𝑓𝑘 ≥ 0 and
𝑔𝑘 = 1. Hence we have obtained
a finite partition of unity. Also, Supp 𝑓𝑘 = Supp 𝑔𝑘 ⊂ 𝑈𝛼𝑘 . To obtain a
partition of unity with the same set of indices, consider functions ℎ𝛼 defined
as 𝑓𝑘 for 𝛼 = 𝛼𝑘 and as 0 otherwise. Note that in both versions the functions
in the partition of zero have compact support (which would not be the case
for a non-compact manifold).
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Example 3.8. A partition of unity for the sphere 𝑆 2 . Use spherical coordinates. One can define non-negative functions 𝑔1 (𝜃) and 𝑔2 (𝜃) such that
𝑔1 (𝜃) = 1 for 0 ≤ 𝜃 ≤ 𝜋/3 and 𝑔1 (𝜃) = 0 for 2𝜋/3 ≤ 𝜃 ≤ 𝜋 and, similarly,
𝑔2 (𝜃) such that 𝑔2 (𝜃) = 1 for 2𝜋/3 ≤ 𝜃 ≤ 𝜋 and 𝑔2 (𝜃) = 0 for 0 ≤ 𝜃 ≤ 𝜋/3.
They are bump functions for the points 𝑁 and 𝑆 respectively. We set
𝑓1 :=
𝑔1
𝑔1 + 𝑔2
and 𝑓2 :=
𝑔2
.
𝑔1 + 𝑔2
Hence (𝑓1 , 𝑓2 ) is a partition of unity subordinate to the open cover (𝑈1 , 𝑈2 )
where 𝑈1 = 𝑆 2 ∖ {𝑆} and 𝑈2 = 𝑆 2 ∖ {𝑁 }.
Corollary 3.1. Let 𝐶0 , 𝐶1 ⊂ 𝑀 be closed subsets such that 𝐶0 ∩ 𝐶1 = ∅.
There is a function 𝑓 ∈ 𝐶 ∞ (𝑀 ) such that 𝑓 ≡ 1 on 𝐶1 and 𝑓 ≡ 0 on 𝐶0 .
Proof. Consider the open cover 𝑀 = 𝑈0 ∪ 𝑈1 , where 𝑈𝛼 := 𝑀 ∖ 𝐶𝛼 and a
subordinate partition of unity 𝑓0 + 𝑓1 = 1. Take 𝑓 := 𝑓0 . By definition,
Supp 𝑓0 ⊂ 𝑈0 = 𝑀 ∖ 𝐶0 . Thus 𝑓 = 0 on 𝐶0 . Similarly, 𝑓1 = 0 on 𝐶1 . Thus
𝑓 = 𝑓0 = 1 − 𝑓1 = 1 on 𝐶1 .
Functions constructed in Corollary 3.1 are called Urysohn functions for
the pair 𝐶0 , 𝐶1 . On a topological space the existence of (continuous) Urysohn
functions is a separation property which is stronger than being Hausdorff.
We are now ready to obtain first applications of the existence of partitions
of unity.
Definition 3.4. A (smooth) map of manifolds 𝑓 : 𝑀 𝑛 → 𝑁 𝑘 is an embedding
if it is injective and all the tangent maps 𝑑𝑓 (𝒙) : 𝑇𝒙 𝑀 → 𝑇𝑓 (𝒙) 𝑁 are also
injective (that is, are monomorphisms of vector spaces).
Theorem 3.8. For any compact manifold 𝑀 𝑛 there is an embedding
𝑓 : 𝑀 → ℝ𝑁 ,
for a sufficiently large 𝑁 .
Proof. Recall the construction of bump functions above. Starting from an
arbitrary atlas for 𝑀 , we can always construct a finite atlas with coordinate
domains 𝑈𝑘 , 𝑘 = 1, . . . , 𝑝, having the following property: there are smaller
open sets 𝑊𝑘 ⊂ 𝑈𝑘 and non-negative functions 𝑔𝑘 such that each 𝑔𝑘 = 1
identically on 𝑊𝑘 and 𝑔𝑘 vanish on 𝑀 ∖ 𝑈𝑘 and the sets 𝑊𝑘 still give a cover
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of 𝑀 . Let 𝑥𝑎𝑘 ∈ 𝐶 ∞ (𝑈𝑘 ) be the coordinate functions on 𝑈𝑘 . Define the
functions 𝑦𝑘𝑎 ∈ 𝐶 ∞ (𝑀 ) as 𝑔𝑘 ⋅ 𝑥𝑎𝑘 on 𝑈𝑘 and as zero on 𝑀 ∖ 𝑈𝑘 . (This is welldefined, compare the construction of bump functions). Define the following
smooth map: 𝑓 : 𝑀 → ℝ𝑝(𝑛+1) , by the formula
𝑓 : 𝑥 7→ (𝑦𝑘𝑎 (𝑥), 𝑔𝑘 (𝑥)).
(2)
This map is injective. Indeed, suppose 𝑓 (𝑥) = 𝑓 (𝑦) for some 𝑥, 𝑦 ∈ 𝑀 . Since
the sets (𝑊𝑘 ) cover 𝑀 , there is some 𝑘 such that 𝑔𝑘 (𝑥) = 1. Hence 𝑔𝑘 (𝑦) = 1.
In particular, it follows that 𝑥 and 𝑦 are in the same coordinate domain 𝑈𝑘
(because 𝑔𝑘 = 0 on 𝑀 ∖ 𝑈𝑘 ). By definition of 𝑦𝑘𝑎 , then all their coordinates
there coincide, and hence 𝑥 = 𝑦. Consider now the Jacobi matrix
( 𝑎
)
∂𝑦𝑘 ∂𝑔𝑘
,
,
∂𝑥𝑏𝑙 ∂𝑥𝑏𝑙
which has 𝑛 rows and (𝑛 + 1)𝑝 columns. For each 𝑥 ∈ 𝑀 , there is an open
∂𝑦 𝑎
∂𝑥𝑎
set 𝑊𝑘 containing 𝑥 where 𝑔𝑘 = 1 identically. Thus ∂𝑥𝑘𝑏 = ∂𝑥𝑏𝑘 = 𝛿𝑏𝑎 (no
𝑘
𝑘
summation over 𝑘!). Hence the rank of the Jacobi matrix is 𝑛.
Thus any compact manifold can be considered as a subspace of a Euclidean space. This is true for non-compact manifolds, too, but the proof for
them is more sophisticated.
3.3
Whitney Theorem
What is the dimension 𝑁 ? Is it possible to minimize it?
We use the following statement (basically, from analysis) without proof:
Lemma 3.4 (Corollary of Sard’s Lemma). A smooth map 𝐹 : 𝑀 𝑛 → 𝑁 𝑚
cannot be surjective if 𝑛 < 𝑚.
Theorem 3.9 (Whitney). A manifold 𝑀 𝑛 of dimension 𝑛 can be embedded
into ℝ𝑁 with 𝑁 = 2𝑛 + 1.
Proof. Suppose that an embedding 𝑀 → ℝ𝑁 is given. Is it possible to
decrease the number 𝑁 ? We can identify 𝑀 with its image in ℝ𝑁 . Consider
the orthogonal projection on the hyperplane 𝑙⊥ ⊂ ℝ𝑁 for some direction
𝑙 ∈ ℝ𝑃 𝑁 −1 . The projection is injective on 𝑀 ⊂ ℝ𝑁 if for any two distinct
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THEODORE VORONOV
DIFFERENTIABLE MANIFOLDS. Fall 2009
points 𝒙 and 𝒚 in 𝑀 , the vector 𝒙 − 𝒚 ∈ ℝ𝑁 is not parallel to 𝑙. (That
is, it does not belong to 𝑙.) In the same way, the projection is injective on
tangent vectors if no non-zero vector tangent to 𝑀 is parallel to 𝑙. Denote
˜ := 𝑀 × 𝑀 ∖ Δ where Δ = {(𝒙, 𝒙)} is the ‘diagonal’. Denote 𝑇 ′ 𝑀 :=
𝑀
𝑇 𝑀 ∖ 𝑀 , i.e., the manifold of all non-zero tangent vectors. Consider the
˜ ∪ 𝑇 ′ 𝑀 . We have a map 𝑀
˜ ∪ 𝑇 ′ 𝑀 → ℝ𝑃 𝑁 −1 sending a pair
disjoint union 𝑀
(𝒙, 𝒚) to the line spanned by 𝒙 − 𝒚 ∈ ℝ𝑁 and a tangent vector 𝒖 ∕= 0 to the
line [𝒖]. It is smooth and we can apply the corollary of Sard’s lemma. This
˜ ∪ 𝑇 ′ 𝑀 ) < dim ℝ𝑃 𝑁 −1 = 𝑁 − 1, i.e.,
map is not onto as long as 2𝑛 = dim(𝑀
𝑁 > 2𝑛 + 1. That means that by applying projections, the dimension of the
ambient space can be reduced to 2𝑛 + 1.
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