We study the geometry and topology of compact submanifolds of Riemannian manifolds with arbitrary codimension that satisfy a pinching condition. The ambient space is not required to be a space form, and the pinching condition involves the length of the second fundamental form and the mean curvature. The results obtained, as well as the methods employed, differ substantially between higher-dimensional and four-dimensional submanifolds. Our approach in higher dimensions relies on results from the theory of Riemannian manifolds with nonnegative isotropic curvature and on the Bochner technique. In contrast, in the four-dimensional case, the analysis also relies critically on concepts from four-dimensional geometry. A key observation is that submanifolds satisfying the pinching condition necessarily have nonnegative isotropic curvature, a notion introduced by Micallef and Moore. The results are sharp and extend previous work by several authors, without imposing any additional assumptions on either the topology or the geometry of the submanifold.
A fundamental problem in differential geometry is to understand the interplay between the geometry and topology of Riemannian manifolds. In the context of submanifold theory, it is particularly interesting to investigate how the geometry and topology of submanifolds of Riemannian manifolds are influenced by pinching conditions involving either intrinsic or extrinsic curvature invariants.
For minimal submanifolds of spheres with a sufficiently pinched second fundamental form, a fundamental result was first established by Simons in his seminal work [29]. Later, Chern, do Carmo, and Kobayashi [9] proved a celebrated rigidity theorem. Their contributions have since inspired a wealth of important developments in the study of pinching conditions for submanifolds of space forms, including, for example, [1,18,24,28,30,31,33,34,35].
In the present paper, we study the geometric and topological rigidity of submanifolds f : M n → ∼ M n+m of Riemannian manifolds, without assuming that the ambient manifold has constant sectional curvature, which satisfy the pointwise pinching condition
Here, the functions M at the point f (x). Moreover, S denotes the squared length of the second fundamental form α f : T M × T M → N f M , which takes values in the normal bundle N f M . The mean curvature is defined by H = ∥H f ∥, where the mean curvature vector field is given by
and tr denotes the trace. A slightly stronger pinching condition than condition ( * ) was studied by Xu and Zhao [37], primarily in the form of a strict inequality holding at every point. Later, Xu and Gu [36] investigated Einstein submanifolds of Riemannian manifolds satisfying the same condition, under the assumption that it holds as a strict inequality at some point. Related results were also obtained in [35].
The main objective of this paper is to study the geometric and topological rigidity of compact submanifolds of Riemannian manifolds satisfying condition ( * ). A key observation, already noted in [37,35], is that submanifolds satisfying the pinching condition ( * ) necessarily have nonnegative isotropic curvature, a notion introduced and studied by Micallef and Moore [21]. Our results show that the pinching condition ( * ) imposes strong restrictions on the topology and geometry of the submanifold. Throughout the paper, all submanifolds under consideration are assumed to be connected.
To state the results, we first recall some definitions. A vector δ in the normal space N f M (x) is called a principal normal of an isometric immersion f : M n → ∼ M n+m at a point x ∈ M n if the tangent subspace
is nontrivial. The dimension of this subspace is called the multiplicity of δ. If the multiplicity is at least two, δ is called a Dupin principal normal.
The relative nullity subspace D f (x) of f at a point x ∈ M n is the kernel of the second fundamental form at x, namely
The dimension of the subspace D f (x) is called the index of relative nullity at x.
The points where f is umbilical are the zeros of the traceless part of the second fundamental form.
The first main result of the paper concerns submanifolds of dimension n ≥ 5 satisfying the pinching condition ( * ) and can be stated as follows.
Theorem 1. Let f : M n → ∼ M n+m , n ≥ 5, be an isometric immersion of a compact Riemannian manifold. Assume that inequality ( * ) is satisfied and that ∼ K f min ≥ 0 at every point of M n . Then one of the following holds:
(i) The manifold M n is diffeomorphic to a spherical space form.
(ii) The universal cover of M n is isometric to a Riemannian product R × N , where N is diffeomorphic to S n-1 and has nonnegative isotropic curvature.
(iii) Equality holds in ( * ) at every point, and one of the following occurs:
(a) The immersion f is totally geodesic, and either M n is isometric to the complex projective space CP n/2 , the quaternionic projective space HP n/4 , the Cayley plane OP 2 (all endowed with their canonical Riemannian metrics), or M n is isometric to the twisted complex projective space CP 2k+1 /Z 2 , where the Z 2 -action arises from an anti-holomorphic involutive isometry with no fixed points.
(b) The manifold M n is flat, f is totally geodesic, and
(c) The manifold M n is a quotient (R 2 × S n-2 (r))/Γ, where Γ is a discrete, fixedpoint-free, cocompact subgroup of the isometry group of the standard Riemannian product R 2 × S n-2 (r), and
Moreover, f has index of relative nullity 2, and δ = nH f /(n -2) is a Dupin principal normal vector field of f with multiplicity n -2 satisfying E δ = D ⊥ f . In particular, if n is even and the second Betti number satisfies β 2 (M n ) > 0, then f is totally geodesic, and M n is isometric to the complex projective space CP n/2 , endowed with the Fubini-Study metric up to scaling of the ambient metric.
The second main result of this paper concerns four-dimensional submanifolds satisfying condition ( * ) that are not covered by Theorem 1. The results obtained, as well as the methods employed, differ substan
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