4.1 Index of a Symmetric Bilinear Form
4.1.1 The evolution of the index of a one-parameter family of
n−(B(t)), wheret7→B(t)is a one parameter family of symmetric bilinear forms on a spaceV.
We make the convention that in this subsectionV will always denote a finite dimensional real vector space:
dim(V)<+∞.
We choose an arbitrary norm inV denoted byk · k; we then define the norm of a bilinear formB ∈B(V)by setting:
kBk= sup
kvk≤1 kwk≤1
|B(v, w)|.
Observe that, since V andB(V) are finite dimensional, then any norm in these spaces induces the same topology.
We will first show that the conditionn−(B)≥k(for some fixedk) is an open condition.
4.1.29. LEMMA. Letk≥0be fixed; the set of symmetric bilinear formsB ∈ Bsym(V)such thatn−(B)≥kis open inBsym(V).
PROOF. LetB ∈ Bsym(V) withn−(B) ≥ k; then, there exists a k-dimen-sionalB-negative subspaceW ⊂ V. Since the unit sphere ofW is compact, we have:
sup
v∈W kvk=1
B(v, v) =c <0;
it now follows directly that if A ∈ Bsym(V) and kA −Bk < |c|/2 then A is
negative definite inW, and thereforen−(A)≥k.
4.1.30. COROLLARY. Letk ≥0be fixed; the set of nondegenerate symmetric bilinear formsB ∈Bsym(V)such thatn−(B) =kis open inBsym(V).
PROOF. IfB ∈ Bsym(V)is nondegenerate andn−(B) = k, thenn+(B) = dim(V)−k(see Corollary 4.1.20); by Lemma 4.1.29, forAin a neighborhood of B inBsym(V)we have n−(A) ≥ kandn+(A) ≥ dim(V)−k, from which we
getn−(A) =kanddgn(A) = 0.
4.1.31. COROLLARY. Lett7→B(t)be a continuous curve inBsym(V)defined in some intervalI ⊂IR; ifB(t)is nondegenerate for allt∈I, thenn−(B(t))and n+(B(t))are constant inI.
PROOF. By Corollary 4.1.30, the set of instantst∈I such thatn−(B(t)) =k is open inI for eachk = 0, . . . ,dim(V)fixed. The conclusion follows from the
connectedness ofI.
Corollary 4.1.31 tells us that the indexn−(B(t))and the co-indexn+(B(t)) can only change when B(t) becomes degenerate; in the next Theorem we show how to compute this change whent7→B(t)is of classC1:
4.1.32. THEOREM. LetB: [t0, t1[→Bsym(V)be a curve of classC1; write N = Ker B(t0)
. Suppose that the bilinear formB0(t0)|N×N is nondegenerate;
then there exists ε > 0 such that for t ∈ ]t0, t0+ε[the bilinear form B(t) is nondegenerate, and the following identities hold:
n+(B(t)) =n+(B(t0)) +n+ B0(t0)|N×N , n−(B(t)) =n−(B(t0)) +n− B0(t0)|N×N
. The proof of Theorem 4.1.32 will follow easily from the following:
4.1.33. LEMMA. LetB: [t0, t1[ → Bsym(V) be a curve of classC1; write N = Ker(B(t0)). If B(t0) is positive semi-definite and B0(t0)|N×N is posi-tive definite, then there exists ε > 0 such that B(t) is positive definite for t ∈ ]t0, t0+ε[.
PROOF. Let W ⊂ V be a subspace complementary to N; it follows from Corollary 4.1.24 thatB(t0)is nondegenerate inW, and from Corollary 4.1.14 that B(t0)is positive definite inW. Choose any norm inV; since the unit sphere ofW is compact, we have:
(4.1.16) inf
w∈W kwk=1
B(t0)(w, w) =c0>0;
similarly, sinceB0(t0)is positive definite inN we have:
(4.1.17) inf
n∈N knk=1
B0(t0)(n, n) =c1>0.
SinceB is continuous, there existsε >0such that kB(t)−B(t0)k ≤ c0
2, t∈[t0, t0+ε[, and it follows from (4.1.16) that:
(4.1.18) inf
w∈W kwk=1
B(t)(w, w)≥ c0
2 >0, t∈[t0, t0+ε[. SinceB is differentiable att0 we can write:
(4.1.19) B(t) =B(t0) + (t−t0)B0(t0) +r(t), with lim
t→t0
r(t) t−t0
= 0,
and then, by possibly choosing a smallerε >0, we get:
(4.1.20) kr(t)k ≤ c1
2(t−t0), t∈[t0, t0+ε[ ; from (4.1.17), (4.1.19) and (4.1.20) it follows:
(4.1.21) inf
n∈N knk=1
B(t)(n, n)≥ c1
2(t−t0), t∈]t0, t0+ε[.
From (4.1.18) and (4.1.21) it follows thatB(t)is positive definite inW and inN fort∈]t0, t0+ε[; takingc3 =kB0(t0)k+c21 we obtain from (4.1.19) and (4.1.20) that:
(4.1.22)
B(t)(w, n)
≤(t−t0)c3, t∈[t0, t0+ε[,
provided thatw ∈W,n ∈N andkwk=knk = 1. By possibly taking a smaller ε >0, putting together (4.1.18), (4.1.21) and (4.1.22) we obtain:
B(t)(w, n)2 ≤(t−t0)2c23< c0c1
4 (t−t0)
≤B(t)(w, w)B(t)(v, v), t∈]t0, t0+ε[, (4.1.23)
for allw∈W,n∈N withkwk=knk= 1; but (4.1.23) implies:
B(t)(w, n)2< B(t)(w, w)B(t)(n, n), t∈]t0, t0+ε[,
for all w ∈ W, n ∈ N non zero. The conclusion follows now from
Proposi-tion 4.1.27.
PROOF OFTHEOREM4.1.32. By Theorem 4.1.10 there exists a decomposi-tionV =V+⊕V−⊕N whereV+andV−are respectively aB(t0)-positive and a B(t0)-negative subspace; similarly, we can writeN = N+⊕N− whereN+is a B0(t0)-positive andN−is aB0(t0)-negative subspace. Obviously:
n+(B(t0)) = dim(V+), n−(B(t0)) = dim(V−), n+ B0(t0)|N×N
= dim(N+), n− B0(t0)|N×N
= dim(N−);
applying Lemma 4.1.33 to the restriction ofB toV+⊕N+ and to the restriction of−BtoV−⊕N−we conclude that there existsε >0such thatB(t)is positive definite in V+ ⊕N+ and negative definite in V−⊕N− for t ∈ ]t0, t0+ε[;the conclusion now follows from Corollary 4.1.7 and from Proposition 4.1.9.
4.1.34. COROLLARY. Ift7→B(t)∈Bsym(V)is a curve of classC1defined in a neighborhood of the instantt0 ∈IRand ifB0(t0)|N×N is nondegenerate, where N = Ker B(t0)
, then forε >0sufficiently small we have:
n+(B(t0+ε))−n+(B(t0−ε)) = sgn B0(t0)|N×N .
PROOF. It follows from Theorem 4.1.32 that forε > 0sufficiently small we have:
(4.1.24) n+(B(t0+ε)) =n+(B(t0)) +n+ B0(t0)|N×N
; applying Theorem 4.1.32 to the curvet7→B(−t)we obtain:
(4.1.25) n+(B(t0−ε)) =n+(B(t0)) +n− B0(t0)|N×N .
The conclusion follows by taking the difference of (4.1.24) and (4.1.25).
We will need a uniform version of Theorem 4.1.32 for technical reasons:
4.1.35. PROPOSITION. Let X be a topological space and let be given a con-tinuous map
X ×[t0, t1[3(λ, t)7−→Bλ(t) =B(λ, t)∈Bsym(V) differentiable int, such that ∂B∂t is also continuous inX ×[t0, t1[.
Write Nλ = Ker Bλ(t0)
; assume thatdim(Nλ) does not depend on λ ∈ X and that Bλ00(t0) = ∂B∂t(λ0, t0) is nondegenerate in Nλ0 for some λ0 ∈ X. Then, there existsε > 0 and a neighborhood Uofλ0 inX such thatB0λ(t0) is nondegenerate onNλand such thatBλ(t)is nondegenerate onV for everyλ∈U and for everyt∈]t0, t0+ε[.
PROOF. We will show first that the general case can be reduced to the case that Nλ does not depend onλ∈ X. To this aim, letk = dim(Nλ), that by hypothesis does not depend onλ. Since the kernel of a bilinear form coincides with the kernel of its associated linear operator, it follows from Proposition 2.4.10 that the map λ 7→ Nλ ∈ Gk(V) is continuous inX; now, using Proposition 2.4.6 we find a continuous mapA:U→GL(V)defined in a neighborhoodUofλ0inX such that for allλ∈U, the isomorphismA(λ)takesNλ0 ontoNλ. Define:
Bλ(t) =A(λ)# Bλ(t)
=Bλ(t) A(λ)·, A(λ)· ,
for allλ∈Uand allt∈[t0, t1[. Then,Ker Bλ(t0)
=Nλ0 for allλ∈U; more-over, the mapB defined inU×[t0, t1[satisfies the hypotheses of the Proposition, and the validity of the thesis forBwill imply the validity of the thesis also forB.
The above argument shows that there is no loss of generality in assuming that:
Ker(Bλ(t0)) =N,
for allλ∈ X. We split the remaining of the proof into two steps.
(1) Suppose thatBλ0(t0)is positive semi-definite and thatBλ0
0(t0)is positive definite inN.
LetW be a subspace complementary toN inV; thenBλ0(t0)is positive definite inW. It follows thatBλ(t0)is positive definite inW and thatBλ0(t0) is positive definite inN for allλin a neighborhoodUofλ0 inX. Observe that, by hypothesis, Ker(Bλ(t0)) = N for all λ ∈ U. Then, for all λ ∈ U, Lemma 4.1.33 gives us the existence of a positive numberε(λ)such that Bλ(t)is positive definite for allt∈]t0, t0+ε(λ)[; we only need to look more closely at the estimates done in the proof of Lemma 4.1.33 to see that it is possible to chooseε > 0 independently ofλ, when λruns in a sufficiently small neighborhood ofλ0inX.
The only estimate that is delicate appears in (4.1.20). Formula (4.1.19) de-fines now a functionrλ(t); for eachλ∈U, we apply the mean value inequality
to the functiont7→σ(t) =Bλ(t)−tBλ0(t0)and we obtain:
kσ(t)−σ(t0)k=krλ(t)k ≤(t−t0) sup
s∈[t0,t]
kσ0(s)k
= (t−t0) sup
s∈[t0,t]
kB0λ(s)−B0λ(t0)k.
With the above estimate it is now easy to get the desired conclusion.
(2) Let us prove the general case.
Keeping in mind that Ker(Bλ(t0)) = N does not depend on λ ∈ X, we repeat the proof of Theorem 4.1.32 replacingB(t0)byBλ(t0),B0(t0)by Bλ0
0(t0)andB(t) byBλ(t); we use step (1) above instead of Lemma 4.1.33 and the proof is completed.
4.1.36. EXAMPLE. Theorem 4.1.32 and its Corollary 4.1.34 do not hold with-out the hypothesis thatB0(t0)be nondegenerate inN = Ker(B(t0)); counterex-amples are easy to produce by considering diagonal matricesB(t) ∈ Bsym(IRn).
A naive analysis of the case in which the bilinear formsB(t)are simultaneously diagonalizable would suggest the conjecture that when B0(t0) is degenerate in Ker(B(t0)) then it would be possible to determine the variation of the co-index ofB(t)when t passes throught0 by using higher order terms on the Taylor ex-pansion ofB(t)|N×N aroundt=t0. The following example show that this is not possible.
Consider the curvesB1, B2:IR→Bsym(IR2)given by:
B1(t) =
1 t t t3
, B2(t) =
1 t2 t2 t3
; we have B1(0) = B2(0) andN = Ker B1(0)
= Ker B2(0)
= {0} ⊕IR.
Observe thatB1(t)|N×N =B2(t)|N×N for allt∈IR, so that the Taylor expansion ofB1 coincides with that of B2 inN; on the other hand, for ε > 0 sufficiently small, we have:
n+(B1(ε))−n+(B1(−ε)) = 1−1 = 0, n+(B2(ε))−n+(B2(−ε)) = 2−1 = 1.
Our next goal is to prove that the basis provided by Sylvester’s Inertia Theorem can be written as a differentiable function of the parametertwhen B(t)depends differentiably ont. Towards this goal, we consider the action of the general linear groupGL(V)in the spaceBsym(V)given by:
(4.1.26)
GL(V)×Bsym(V)3(T, B)7−→T#(B) =B(T−1·, T−1·)∈Bsym(V);
it follows from Sylvester’s Inertia Theorem (Theorem 4.1.10) that the orbits of this action are the sets:
Bp,qsym(V) ={B ∈Bsym(V) :n+(B) =p, n−(B) =q},
withp+q = 0,1, . . . ,dim(V). Moreover, forpandqfixed, the sets {B ∈Bsym(V) :n+(B)≥p, n−(B)≥q} and {B ∈Bsym(V) :n+(B)≤p, n−(B)≤q}
are respective an open and a closed subset ofBsym(V), by Lemma 4.1.29. It fol-lows that the setBp,qsym(V)is locally closed in Bsym(V). From these observation we deduce the following
4.1.37. LEMMA. The setBp,qsym(V) is a connected embedded submanifold of Bsym(V)for any integersp, q≥0withp+q = 0,1, . . . ,dim(V).
PROOF. The fact thatBp,qsym(V)is an embedded submanifold ofBsym(V) fol-lows from Theorem 2.1.12. The connectedness ofBp,qsym(V)follows from the fact that the restriction of the action (4.1.26) toGL+(V)is still transitive inBp,qsym(V);
this last statement follows from the fact that, once an orientation has been fixed in V, the basis (bi)ni=1 given by the Sylvester’s Inertia Theorem can be chosen positively oriented (possibly replacingb1with−b1).
4.1.38. COROLLARY. The set of nondegenerate symmetric bilinear forms in V is an open subset ofBsym(V)whose (arc-)connected components are the sets Bk,n−ksym (V),k= 0,1, . . . , n, wheren= dim(V).
PROOF. It follows from Corollary 4.1.30 and Lemma 4.1.37.
We finally obtain the desired extension of Sylvester’s Inertia Theorem:
4.1.39. PROPOSITION. Given a curveB: [a, b]→Bsym(V)of classCk(0≤ k ≤ +∞) such that the integers n−(B(t))andn+(B(t))do not depend ont ∈ [a, b], then there exist mapsbi: [a, b]→ V of classCk,i= 1, . . . , n, such that for eacht∈[a, b]the vectors(bi(t))ni=1 form a basis ofV in whichB(t)assumes the canonical form (4.1.7).
PROOF. Letpandq be such thatn+(B(t)) = p, n−(B(t)) = q for allt ∈ [a, b]; keeping in mind the transitive action (4.1.26) of GL(V) on Bp,qsym(V), it follows from Corollary 2.1.15 that, forB0∈Bp,qsym(V)fixed, the map
GL(V)3T 7−→T#(B0) =B0(T−1·, T−1·)∈Bp,qsym(V)
is a differentiable fibration. It follows from Remark 2.1.18 that there exists a map T: [a, b] → GL(V)of class Ck such that T(t)#(B0) = B(t)for allt ∈ [a, b].
Choosing a basis (bi)ni=1 ofV with respect to which B0 has the canonical form (4.1.7), we definebi(t) =T(t)·bifori= 1, . . . , nandt∈[a, b]. This concludes
the proof.