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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 9, No. 3, 2016, 314-321

ISSN 1307-5543 – www.ejpam.com

Perfect Morse Function on SO(n)

Mehmet Solgun

Bilecik Seyh Edebali University, Faculty of Sciences and Arts, Department of Mathematics, Bilecik,

TURKEY

Abstract. In this work, we define a Morse function on SO(n) and show that this function is indeed a

perfect Morse function.

2010 Mathematics Subject Classifications: 57R70, 58E05

Key Words and Phrases: SO(n), Morse functions, Perfect Morse functions.

1. Introduction

The main point of Morse Theory, which was introduced in [6], is investigating the relation

between shape of a smooth manifold M and critical points of a specific real-valued function

f : M → R, that is called Morse function. [5] and [4] are two of main sources about this subject,

so mostly we will use their beautiful tools for defining a Morse function on SO(n). Also, we will

refer [2] to use homological properties and to determine the Poincaré polynomial of SO(n).

Perfect Morse functions are widely studied in [7], that is one of our inspiration to show that

the function, we defined, is also perfect.

2. Preliminaries

In this section, we give some definitions and theorems which will be used in this paper.

Definition 1. Let M be an n-dimensional smooth manifold and f : M → R be a smooth function.

A point p0 ∈ M is said to be a critical point of M if we have

∂ f

∂ x1

= 0,
∂ f

∂ x2

= 0, . . . ,
∂ f

∂ xn

= 0 (1)

with respect to a coordinate system {x1, x2, . . . , xn} around p0.

Email address: mehmet.solgun@bilecik.edu.tr

http://www.ejpam.com 314 c© 2016 EJPAM All rights reserved.



M. Solgun / Eur. J. Pure Appl. Math, 9 (2016), 314-321 315

A point c ∈ R is said to be a critical value of f : M → R, if f (p0) = c for a critical point p0

of f .

Definition 2. Let p0 be a critical point of the function f : M → R. The Hessian of f at he point

p0 is the n× n matrix

H f (p0) =









∂ 2 f

∂ x1
2 (p0) · · · ∂ 2 f

∂ x1∂ xn
(p0)

...
. . .

...
∂ 2 f

∂ xn∂ x1
(p0) · · ·

∂ 2 f

∂ xn
2 (p0)









(2)

Since
∂ 2 f

∂ x i∂ x j
(p0) =

∂ 2 f

∂ x j∂ x i
(p0), the Hessian of f is a symmetric matrix.

Let p0 be a critical point of f and c0 ∈ R such that f (p0) = c0. Then, c0 is said to be a

critical value of f . If p0 is a regular point of f , then c0 is said to be a regular value of f .

If a is a regular value of f , it can be shown that the set f −1(a) = {p ∈ M | f (p) = a} is an

n− 1 dimensional manifold [1].

Definition 3. A critical point of a function f : M → R is called "non-degenerate point of f " if

detH f (p0) 6= 0. Otherwise, it is called "degenerate critical point".

Lemma 1. Let p0 be a critical point of a smooth function

f : M → R, (U ,ϕ = (x1, . . . , xn)), (V,ψ= (X1, . . . , Xn))

be two charts of p0, and H f (p0),H f (p0) be the Hessians of f at p0, using the charts (U ,ϕ),

(V,ψ) respectively. Then the following holds:

H f (p0) = J(p0)
t H f (p0)J(p0) (3)

where J(p0) is the Jacobian matrix for the given coordinate transformation, defined by

J(p0) =









∂ x1

∂ X1
(p0) · · ·

∂ x1

∂ Xn
(p0)

...
. . .

...
∂ xn

∂ X1
(p0) · · ·

∂ xn

∂ Xn
(p0)









(4)

and the matrix J(p0)
t is the transpose of J(p0).

For a critical point p0, non-degeneracy does not depend on the choice of charts around p0.

The same argument is also true for degenerate critical points. In fact we have

H f (p0) = J(p0)
t H f (p0)J(p0)

by the previous lemma, and hence

detH f (p0) = detJ(p0)
t detH f (p0)detJ(p0) (5)



M. Solgun / Eur. J. Pure Appl. Math, 9 (2016), 314-321 316

by using determinant function on both sides. On the other hand, the determinant of the

Jacobian matrix is non-zero. So the statement "detH f (p0) 6= 0" and "detH f (p0) 6= 0" are

equivalent. In other words,

detH f (p0) 6= 0⇔ detH f (p0) 6= 0.

Now a function f : M → R is called a Morse function if any critical point of f is non-degenerate.

From now on, we only consider a Morse function f .

Now, we introduce Morse lemma on manifolds.

Theorem 1 (The Morse Lemma). Let M be an n-dimensional smooth manifold and p0 be a non-

degenerate critical point of a Morse function f : M → R. Then, there exists a local coordinate

system (X1, X2, . . . , Xn) around p0 such that the coordinate representation of f has the following

form:

f = −X 2
1 − X 2

2 . . .− X 2
λ + X 2

λ+1
+ . . .+ X 2

n + c (6)

where c = f (p0) and p0 corresponds to the origin (0,0, . . . , 0).

One may refer to [5] for the proof.

The number λ of minus signs in the equation (6) is the number of negative diagonal entries

of the matrix H f (p0) after diagonalization. By Sylvester’s law, λ does not depend on how

H f (p0) is diagonalized. So, λ is determined by f and p0. The number λ is called "the index of

the non-degenerate critical point p0". Obviously, λ is an integer between 0 and n. Note that,

(i) A non-degenerate critical point is isolated.

(ii) A Morse function on a compact manifold has only finitely many critical points [5].

3. A Morse Function on SO(n)

In this section, we will define a Morse function on SO(n).

The set of all n× n orthogonal matrices, O(n) = {A= (ai j) ∈ Mn(R) : AAt = In} is a group

with matrix multiplication. From the definition of O(n),

detA= ±1, for any A∈ O(n).

An orthogonal matrix with determinant 1 is called rotation matrix and the set of this kind

of matrices is also a group, called special orthogonal group and denoted by SO(n). On the other

hand, let Sn(R) denote the set of symmetric n × n matrices. Since each symmetric matrix is

uniquely determined by its entries on and above the main diagonal, that is a linear subspace

of Mn(R) of dimension n(n+ 1)/2.

Now we define a function ϕ : GLn(R) −→ Sn(R) by

ϕ(A) := AtA.

Then, the identity matrix In is a regular value of ϕ [3].



M. Solgun / Eur. J. Pure Appl. Math, 9 (2016), 314-321 317

Let C ∈ Sn(R) with entries ci , with 0≤ c1 < c2 < . . .< cn fixed real numbers and

fC : SO(n)→ R be given by,

fC(A) :=< C ,A>= c1 x11 + c2 x22 + . . .+ cn xnn, (7)

where A= (x i j) ∈ SO(n).

Obviously, fC is a smooth function. Now, we will determine its critical points.

Lemma 2. The critical points of the function fC defined above are:











±1 0 · · · 0

0 ±1
. . .

...
...

. . .
. . . 0

0 · · · 0 ±1











(8)

Proof. Let A be a critical point of fC . Then the derivative of fC at A must be zero. Consider

the matrix given by a rotation of first and second coordinate B12(θ ) defined by

B12(θ ) =















cosθ −sinθ 0 · · · 0

sinθ cosθ 0 · · · 0

0 0 1
...

...
...

. . .
...

0 0 · · · 0 1















.

Then, AB12(θ ) ∈ SO(n) and the matrix B12(θ ) forms a curve on SO(n). Moreover, B12(θ ) = A

for θ = 0.

By the definition of fC , and after computing the matrix product, we have

fC(AB12(θ )) = c1(x11cosθ + x12sinθ ) + c2(−x21sinθ + x22cosθ ) + c3 x33 + . . .+ cn xnn. (9)

By differentiating f in the direction of the velocity vector d
dθ AB12(θ )|θ=0 of the curve

AB12(θ ) at A, we have
d

dθ
fC(AB12(θ ))|θ=0 = c1 x12 − c2 x21 (10)

and
d

dθ
fC(B12(θ )A)|θ=0 = −c1 x21 + c2 x12. (11)

However, by the assumption that A is a critical point of fC , we require these derivatives to

be zero. i.e.

c1 x12 − c2 x12 = 0

−c1 x21 + c2 x12 = 0

Solving this system for x12, x21 gives x12 = x21 = 0. We can carry out the similar calculation for

Bi j(θ ) with i < j, where Bi j(θ ) is with the entries: (i, i) = cosθ , (i, j) = −sinθ , ( j, i) = sinθ



M. Solgun / Eur. J. Pure Appl. Math, 9 (2016), 314-321 318

and ( j, j) = cosθ . Thus, for the matrix A, x i j = 0 whenever i 6= j. So, that is, a critical point of

fC is a diagonal matrix. On the other hand A∈ SO(n), so we have AAt = In. So each entry on

the main diagonal of A must be ±1.

Conversely, let A be a matrix in the form (8). In order to check that A is a critical point, we

need to compute the derivative of fC . If we could find n(n− 1)/2 curves Ci going through A

with velocity vector at A and linearly independent from each other. Since the velocity vector

of Ci at A plays a role of a local coordinate of A, we only need to check that the derivative of

fC(Ci) vanishes to see that D f (A) = 0. Now, the claim is the curves Ci ’s are in fact ABi jθ ’s

defined above. Let εi = Aii where Aii is the i-th diagonal entry of A(εi = ±1). Then, the

derivative of the matrix ABi j(θ ) at A is (we did for the case B12, but it is same for other indices

with i < j),

d

dθ
AB12(θ )|θ=0 =















0 −ε1 0 · · · 0

ε2 0 0 · · · 0

0 0
... 0

...
...

...

0 0 · · · 0















This matrix is regarded as a vector in Rn2

. By considering all 1 ≤ i ≤ j ≤ n, these matrices

(vectors) form a basis for the tangent space TASO(n).

So, for a given matrix A in the form (8), it is easy to compute that, the derivative of fC at

A is zero. This means nothing but A is a critical point of fC .

After now, we know the coordinate system of SO(n) and the critical points of the the given

function fC . It is straightforward to compute the Hessian of fC at A. Suppose that A is a critical

matrix with diagonal entries Aii = εi = ±1. Then, we want to compute

∂ 2

∂ θ∂ ϕ
fC(ABαβ (θ )Bγδ(ϕ))|θ=0,ϕ=0.

Notice that is linear ABαβ (θ )Bγδ(ϕ) is linear in θ and in ϕ, and fC is a linear function. Thus,

we can bring the derivative inside fC . So,

∂ 2

∂ θ∂ ϕ
fC(ABαβ(θ )Bγδ(ϕ))|θ=0,ϕ=0 = fC(A

d

dθ
Bαβ (θ )|θ=0

d

dϕ
Bγδ(ϕ)|ϕ=0

=

¨

−cαεα − cβεβ if α= γ,β = δ

0 otherwise

This calculation becomes easier if we consider the matrix multiplication ci j =
∑

k aik bk j .

The calculation above shows that the Hessian matrix is diagonal. Since cα 6= cβ for α 6= β ,

the entries on the diagonal are non-zero. Therefore, A is a non-degenerate critical point of fC ,

meaning that fC is a Morse function on SO(n).

Assume that the subscripts i of the diagonal entries εi of A, 1≤ i ≤ n, with εi = 1 are

i1, i2, . . . , im



M. Solgun / Eur. J. Pure Appl. Math, 9 (2016), 314-321 319

in ascending order. Then the index of the critical point A ( the number of minus signs on the

diagonal of Hessian) is

(i1 − 1) + (i2 − 1) + . . .+ (im − 1).

And the index is 0 if all εi ’s are -1. Also, the critical value at the critical point is

2(ci1 + ci2 + · · ·+ cim)−
n
∑

i=0

ci .

Considering that detA= 1, there are 2n−1 critical points [4].

4. Perfect Morse Functions

First, we will give the basic notions.

Definition 4. The Poincaré polynomial of the n-dimensional manifold M is defined to be

PM (t) =

n
∑

k=0

bk(M)t
k (12)

where bk(M) is the k-th Betti number of M.

Definition 5. Let f : M → R be a Morse function. Then, the Morse polynomial of f is defined to

be

Pf (t) =

n
∑

k=0

µk tk (13)

where µk is the number of critical points of f of index k.

Theorem 2 (The Morse Inequality). Let f : M → R be a Morse function on a smooth manifold

M. Then, there exists a polynomial R(t) with non-negative integer coefficients such that

Pf (t) = PM (t) + (1+ t)R(t).

One may refer to [7] for proof.

A Morse function f : M → R is called a perfect Morse function if Pf (t) = PM (t) [7].

Now, we show that the function fC on SO(n) defined in the previous section is also a perfect

Morse function.

Theorem 3. The function

fC : SO(n)→ R, fC(A) := 〈C ,A〉

is a perfect Morse function where C ∈ Sn(R).



M. Solgun / Eur. J. Pure Appl. Math, 9 (2016), 314-321 320

Proof. First we show that the Morse polynomial is,

PfC
(t) = (1+ t)(1+ t2) · · · (1+ tn−1). (14)

We use induction method. For making it easier, we label the function fC with n as

fC n : SO(n)→ R.

Trivially, for n= 1, PfC 1
(t) = 1 and for n= 2, PfC 2

(t) = 1+ t. Assume that

PfC n
(t) = (1+ t)(1+ t2) + . . .+ (1+ tn−1). Then, we need to show that PfC n+1

(t) satisfies the

form (14).

We may consider that SO(n+ 1) gets all the critical points from SO(n) with extra bottom

entry ((n+ 1)-th diagonal entry), which is either +1 or -1. Say the set of all these points are

C+
n+1

and C−
n+1

respectively.

Let A∈ C−
n+1

. Then we have Ã∈ O(n) such that, A is the matrix à with extra bottom entry

-1. Then, by the definition of index, we obtain

ind(A) = ind(Ã).

Thus, for the elements of C−
n+1

the equation (14) holds. Let A∈ C+
n+1

. Then we have Ã∈ SO(n)

such that, A is the matrix with à with the bottom entry +1. Thus, by the definition of index,

we obtain

ind(A) = ind(Ã) + n.

So, by the definition of Morse polynomial, we gain

PfC n+1
(t) = PfC n

(t)(1+ tn) = (1+ t)(1+ t2) . . . (1+ tn−1)(1+ tn). (15)

Now, we find out the Poincaré polynomial of SO(n). The graded abelian group

H∗(SO(n),Z2) is isomorphic to the graded group coming from the exterior algebra [2]

∧Z2
[e1, e2, . . . , en−1].

Let say A(n) = ∧Z2
[e1, e2, . . . , en−1] where the degree of ei , |ei |= i. Then, we obtain

|ei1
∧ ei2
∧ . . .∧ eik

|=
k
∑

j=1

|ei j
|=

k
∑

j=1

i j .

If we define a(n)k = dimZ2(A(n)k), then by the result in [2], a(n)k is nothing but the k-th

Betti number of SO(n). Hence, the polynomial

P(A(n)) =

∞
∑

i=0

a(n)i t
i

is the Poincaré polynomial of SO(n).

Now, our claim is that The Poincaré polynomial of SO(n) is

P(A(n)) = (1+ t)(1+ t2) . . . (1+ tn−1).



REFERENCES 321

Let B(A(n)) be the basis of A(n). For instance, B(A(1)) = trivial, B(A(2)) = {1, e1},
B(A(3)) = {1, e1, e2, e1 ∧ e2} etc.

In this sense, we obtain

B(A(n+ 1)) = (B(A(n))∧ en)⊔ B(A(n)).

We use induction method. Indeed, here we have very similar arguments with the previous

claim. The variable en has the same role with " the extra bottom entry ±1". Then, we have the

polynomial P(A(n)) =
∑

b∈B(A(n)) a(n)b t |b|. Trivially, P(A(1)) = 1 and P(A(2)) = 1+ t. By the

induction hypothesis, assume that

P(A(n)) =
∑

b∈B(A(n))

a(n)b t |b| = (1+ t)(1+ t2) · · · (1+ tn−1).

For the polynomial P(A(n+1)), pick an element b ∈ B(A(n+1)). Then, b is in either B(A(n)) or

B(A(n))∧en. For b ∈ B(A(n+1)), trivially, P(A(n+1)) has the desired form. If b ∈ B(A(n))∧en,

then by the definition of degree, there is b̃ ∈ B(A(n)) such that |b| = |b̃| + n. Thus, by the

definition of P(A(n)), we obtain

P(A(n+ 1)) = (1+ t)(1+ t2) · · · (1+ tn) (16)

which completes the proof.

Thereby, we have shown that, for the given Morse function fC : SO(n)→ R, PM (t) = PfC
(t),

meaning that fC is a perfect Morse function.

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[2] A. Hatcher. Algebraic Topology. Cambridge University Press, New York, 3rd edition, 2002.

[3] J.M. Lee. Introduction to Smooth Manifolds. Springer, New York, 2nd edition, 2012.

[4] Y. Matsumoto. An Introduction to Morse Theory. American Mathematical Society, Transla-

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[5] J.W. Milnor. Morse Theory. Princeton University Press, New Jersey, USA, 1963.

[6] J. Morse. The foundations of a theory of the calculus of variations in the large in m-space.

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