Lemme B.2. Soient (M, g) une variété riemannnienne compacte et (Sm, h
m) la sphère standard. Sur la variété M × Smmunie de la métrique produit g × h
m, on considère le groupe
G = IM× O(r1) × O(r2),
où r1 ≥ r2et r1+r2= m+1. Alors pour tout x0= (y0, 0Rr1, z0) avec y0∈ M, et z0∈ Sr2−1, OG
x0est une orbite de dimension et de volume minimal parmi les G-orbites et le sous
groupe normal de G noté H = IM× Ir1× O(r2) permet le passage au quotient sur un
voisinage Ox0,δ= {x ∈ M×S m, d g×hm(x, O G x0)< δ} de O G x0. On noteπH: Ox0,δ→ Ox0,δ/H
la projection canonique, ˜g la métrique quotient induite par g et ¯x0= πH(OHx0). Alors
S cal˜g( ¯x0) ≥ S calg(y0) + r1(r1− 1) Démonstration du lemme : Ox0,δ= {x ∈ M × S m, d g×hm(x, O G x0)< δ} est un ouvert de M × S mcontenant l’orbite OGx0= {y0} × {0Rr1} × Sr2−1où y0∈ M. Il existe donc un ouvert O1de M contenant y0et
un ouvert O2de Smcontenant {ORr1} × Sr2−1tels que
et on a en notant H′= Ir1× O(r2)
(O1× O2)/H = O1 × (O2/ H′)
avec pour métrique quotient ˜g = g × ˜hmoù ˜hmest la métrique quotient induite par hm
sur Sm/H′. D’autre part comme OG x0= O H x0 ¯x0= πH(OHx0) = πH {y0} × {0Rr1} × Sr2−1 = {y0} × p {0r1} × S r2−1= {y 0} × {t0}
où p : Sm → Sm/H′ est la projection canonique et t
0 = p {0r1} × S r2−1 ∈ O 2/H. Ainsi,
S cal˜g( ¯x0) = S calgטhm({y0} × {t0})
= S calg(y0) + S cal˜hm(t0)
Or t0 ∈ O2/H′ ⊂ Sm/H′et d’après le lemme B.1, puisque Sm/H′est de dimension m − r2+ 1 = r1et que la courbure sectionnelle de la sphère vaut +1, on a : S cal˜hm(t0) ≥
r1(r1− 1). Ainsi :
S cal˜g( ¯x0) ≥ S calg(y0) + r1(r1− 1),
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