Splitting Theorem Research Articles

The equivalence of smooth and synthetic notions of timelike sectional curvature bounds

Timelike sectional curvature bounds play an important role in spacetime geometry, both for the understanding of classical smooth spacetimes and for the study of Lorentzian (pre-)length spaces introduced by Kunzinger and Sämann [Ann. Global Anal. Geom. 54 (2018), pp. 399-447]. In the smooth setting, a bound on the sectional curvature of timelike planes can be formulated via the Riemann curvature tensor. In the synthetic setting, bounds are formulated by comparing various geometric configurations to the corresponding ones in constant curvature spaces. The first link between these notions in the Lorentzian context was established by Harris [Indiana Univ. Math. J. 31 (1982), pp. 289–308], which was instrumental in the proof of powerful results in spacetime geometry (see Beem et al. [Toponogov splitting theorem for Lorentzian manifolds, Springer, Berlin, 1985; J. Differential Geom. 22 (1985), pp. 29–42]; Galloway and Ling [Gen. Relativity Gravitation 50 (2018), p. 7]). For general semi-Riemannian manifolds, the equivalence between sectional curvature bounds and synthetic bounds was established by Alexander and Bishop [Comm. Anal. Geom. 16 (2008), pp. 251–282]; however in this approach the sectional curvatures of both timelike and spacelike planes have to be considered. In this article, we fill a gap in the literature by proving the full equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes. As an essential tool, we establish Hessian comparison for the time separation and signed distance functions.

On collection schemes and Gaifman's splitting theorem

AbstractWe study model theoretic characterizations of various collection schemes over from the viewpoint of Gaifman's splitting theorem. Among other things, we prove that for any and , the following are equivalent: satisfies the collection scheme for formulas. For any , if , and , then and . For any , if , then . Here, is the unique satisfying . We also investigate strong collection schemes and parameter‐free collection schemes from the similar perspective.

Nijenhuis operators with a unity and F$F$‐manifolds

AbstractThe core object of this paper is a pair , where is a Nijenhuis operator and is a vector field satisfying a specific Lie derivative condition, that is, . Our research unfolds in two parts. In the first part, we establish a splitting theorem for Nijenhuis operators with a unity, offering an effective reduction of their study to cases where has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for ‐regular Nijenhuis operators with a unity around algebraically generic points, along with seminormal forms for dimensions 2 and 3. In the second part, we establish the relationship between Nijenhuis operators with a unity and ‐manifolds. Specifically, we prove that the class of regular ‐manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding ‐manifolds around singularities.

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

Abstract The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl -regular Nijenhuis operators. We prove the splitting theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl -regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl -regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl -regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e. its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.

Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum.

Eigenvalue lower bounds and splitting for modified Ricci flow

Abstract We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.

Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

We prove splitting theorems for mean convex open subsets in Riemannian curvature-dimension (RCD) spaces that extend results by Kasue et al. for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel’s theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable with respect to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.

Normal forms of [formula omitted]-graded Q-manifolds

Following recent results of A.K. and V.S. on Z-graded manifolds, we give several local and global normal forms results for Q-structures on those, i.e. for differential graded manifolds. In particular, we explain in which sense their relevant structures are concentrated along the zero-locus of their curvatures, especially when the negative-part is of Koszul-Tate type. We also give a local splitting theorem.

Equivariant [formula omitted]-modules for the cyclic group C2

For the cyclic group C2 we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring Z/ℓ_, for ℓ a prime. This is fairly simple for ℓ odd, but for ℓ=2 depends on a new splitting theorem. As corollaries of the splitting theorem we compute the associated Picard group and the Balmer spectrum for compact objects in the derived category, and we obtain a complete classification of finite modules over the C2-equivariant Eilenberg–MacLane spectrum HZ/2_. We also use the splitting theorem to give new and illuminating proofs of some facts about RO(C2)-graded Bredon cohomology, namely Kronholm's freeness theorem [12,11] and the structure theorem of C. May [13].

Cohomological χ–independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $\chi$-independence conjecture by Bousseau for $\mathbb{P}^2$, and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces. For the proof, we combine a generalized version of Ng\^o's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.

Victor Percy Snaith, 1944–2021

AbstractVictor Snaith was born in Colchester, Essex, on 15 March 1944 and passed away in Bristol on 3 July 2021 at the age of 77. He completed his PhD at the University of Warwick in 1969 under the supervision of Luke Hodgkin. He held faculty positions in Canada, at the University of Western Ontario (1976–88) and McMaster University (1988–98), and became a leading figure in the Canadian mathematical research community during that time. He was named a Fellow of the Royal Society of Canada in 1984, and held the first Britton Professorship at McMaster University. He returned to the United Kingdom in 1998, to professorships at the University of Southampton (1998–2004) and the University of Sheffield (2004–2009) from where he retired in 2009. He was an active member of the Heilbronn Institute from 2006 until the time of his passing. Snaith made significant research contributions in various areas, including the Snaith splitting theorem in stable homotopy theory, the introduction of the Bott periodic form of algebraic K‐theory, and the explicit Brauer induction theorem with its applications in algebraic number theory. He is survived by his wife Carolyn, son Dan, and daughters Anna and Nina.

Normal forms for Dirac–Jacobi bundles and splitting theorems for Jacobi structures

The aim of this paper is to prove a normal form Theorem for Dirac–Jacobi bundles using a recent techniques of Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As another application we provide an alternative proof of the splitting theorem of homogeneous Poisson structures.

A generalization of Kruskal’s theorem on tensor decomposition

Abstract Kruskal’s theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a ‘splitting theorem’ for sets of product tensors, in which the k-rank condition of Kruskal’s theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e., is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal’s theorem. While several extensions of Kruskal’s theorem are already present in the literature, all of these use Kruskal’s original permutation lemma and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester’s matrix rank inequality to tensors. We also prove novel uniqueness results for nonrank tensor decompositions.

Geometry of weighted Lorentz–Finsler manifolds II: A splitting theorem

We show an analog of the Lorentzian splitting theorem for weighted Lorentz–Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the [Formula: see text]-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar–Wylie into a single framework.

Cheeger–Gromoll splitting theorem for groups

We study a notion of curvature for finitely generated groups which serves as a role of Ricci curvature for Riemannian manifolds. We prove an analog of Cheeger-Gromoll splitting theorem. As a consequence, we give a geometric characterization of virtually abelian groups. We also explore the relation between this notion of curvature and the growth of groups.

A global Weinstein splitting theorem for holomorphic Poisson manifolds

We prove that if a compact K\"ahler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite \'etale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville's conjecture on the structure of compact K\"ahler manifolds with split tangent bundle.

A note on local minimizers of energy on complete manifolds

In this paper, we study the geometric rigidity of complete Riemannian manifolds admitting local minimizers of energy functionals. More precisely, assuming the existence of a non-trivial local minimizer and under suitable assumptions, a Riemannian manifold under consideration must be a product manifold furnished with a warped metric. Secondly, under similar hypotheses, we deduce a geometrical splitting in the same fashion as in the Cheeger-Gromoll splitting theorem and we also get information about local minimizers.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

In this paper, based on the slice regular quaternionic functions and spectral theory on the S-spectrum for quaternionic operators, firstly, the exponential growth bound of quaternionic operator semigroups is studied and the generalized Hille Lemma is obtained through introducing Bochner and Pettis integrals in quaternionic Banach space. Secondly, we study the quaternionic bisemigroups and the direct sum decomposition of quaternionic Banach space, then we introduce the notion of the quaternionic exponentially dichotomous operator and obtain its integral representation formula. It is crucial to note that the quaternionic operators we consider do not necessarily commute. Meanwhile, through establishing the S-spectral splitting theorem, the characterizations of quaternionic exponentially dichotomous operators are demonstrated. Moreover, the slice regular quaternionic bisemigroups generated by the quaternionic bisectorial operator are studied. Thirdly, by introducing the slice quaternionic Banach algebra and multiplicative quaternionic linear functionals, the perturbation invariance of exponential dichotomy for the quaternionic operators is explored. Under the noncommutative setting, our obtained results are essentially different from their analogues in the complex setting. As applications, we consider the Cauchy problems of quaternionic non-homogeneous evolution equations involving the quaternionic semigroup generators and exponentially dichotomous quaternionic operators. Additionally, the slice exponential dichotomy of quaternionic operators is characterized via the corresponding Cauchy problem.

Complex Dirac Structures: Invariants and Local Structure

We study complex Dirac structures, that is, Dirac structures in the complexified generalized tangent bundle. These include presymplectic foliations, transverse holomorphic structures, CR-related geometries and generalized complex structures. We introduce two invariants, the order and the (normalized) type. We show that, together with the real index, they allow us to obtain a pointwise classification of complex Dirac structures. For constant order, we prove the existence of an underlying real Dirac structure, which generalizes the Poisson structure associated to a generalized complex structure. For constant real index and order, we prove a splitting theorem, which gives a local description in terms of a presymplectic leaf and a small transversal.

Polar foliations on symmetric spaces and mean curvature flow

Abstract In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos, Isoparametric submanifolds and a Chevalley-type restriction theorem, Integrable systems, geometry, and topology, American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng, Ancient solutions to mean curvature flow for isoparametric submanifolds, Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres.