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Integral submanifolds of \(r\)-contact manifolds. (English) Zbl 1148.53015

Let \((M^{2n+r};\eta_1 ,\dots ,\eta_r)\) be an \(r\)-contact manifold. An integral submanifold of this \(r\)-contact manifold is a submanifold \(M^s\) of \(M^{2n+r}\) such that any vector tangent to \(M^s\) belongs to the distribution \(\mathcal{D}\) defined by \(\{ \eta_1 =0,\dots ,\eta_r =0\}\). In this case \(s\leq n\). When \(s=n\) the integral submanifold is called a Legendrian submanifold.
The author finds some properties of integral submanifolds. The immersion of integral submanifolds into the ambient space endowed with a compatible metric \(f\)-structure, is studied. He gives conditions (in terms of sectional curvature, Ricci curvature or scalar curvature) which ensure that a minimal Legendrian submanifold of an \(\mathcal{S}\)-space form is totally geodesic.

MSC:

53C12 Foliations (differential geometric aspects)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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