We show that the systolic area of every nonpositively curved closed surface M other than the torus is at least 1, with equality if and only if M is isometric to a square flat Klein bottle. The proof focuses on the Klein-double 4ℝℙ^2 and exploits Weil’s isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.
We investigate complex geometric properties of compact complex solvmanifolds endowed with left-invariant complex structures, focusing on the case where these manifolds admit a complex parallelizable solvmanifold as a finite covering. While our previous paper examined differences among complex solvmanifolds admitting a common complex parallelizable solvmanifold as a finite covering, the present paper is devoted to the study of their common geometric properties.
In this paper, we investigate the properties of the normal Jacobi operator of a real hypersurface in non-flat quaternionic space forms, focused on cyclic parallelism and Codazzi type properties. We classify real hypersurfaces that satisfy these properties. As a solution of the classification problem for such real hypersurfaces, we present the following results: Firstly, we prove that a real hypersurface satisfying a certain condition for the normal Jacobi operator is curvature-adapted. Based on this fact, we further show that in the quaternionic hyperbolic space, the shape operator A of M commutes with the three structure tensors ϕ _ν , ν =1,2,3 , of M on the maximal quaternionic subbundle 𝔇 .
We introduce the concept of chiral geometric operators and use Gilkey’s invariance theory to prove the local index theorem for these operators. In other words, we demonstrate that the supertrace of the heat kernel of a given geometric operator converges as time approaches zero and that this limit is the Chern–Weil form of the Atiyah–Singer integrand. In addition to classical Dirac-type operators that appear in geometry, chiral geometric operators include all higher Dirac operators. This includes in particular the Rarita–Schwinger operator. We also construct a new class of such operators on four-manifolds called higher signature operators.
We analyze the geometry of the field equations of Cotton gravity (for a quite general energy-momentum tensor) on a static space-time. In particular, we describe the local structure of the spatial Riemannian factor. This structure, that we call Cotton-φ-perfect fluid (C-φ-PF, for shorts) is a generalization to the regime of Cotton Gravity of the recently introduced notion of φ-static perfect fluid space-time (φ-SPFST). After discussing the variational origin of this system, we provide sufficient conditions for a C-φ-PF to reduce to a φ-SPFST. We also study the geometry of the level sets of the lapse function f and we provide a rigidity result for C-φ-PFs under some curvature conditions. The role that Codazzi tensors hold in this theory is highlighted.
In this paper, we study conformal actions on the Euclidean sphere 𝕊^n , focusing on the case where the action induces an open orbit. Under this condition, we determine the Lie algebras of the acting Lie groups up to conjugacy within the orthogonal group O(n, 1). Furthermore, we classify these actions up to orbit equivalence and specify the resulting orbits up to conformal equivalence.
We prove vector-valued boundedness of (suitable) Calderón-Zygmund operators and of the (truncated) Hardy–Littlewood maximal function on a connected locally doubling metric measure space.
In this work, we revisit quasi-Sasakian structures in dimension three and examine how these structures interact with the foliation generated by the Reeb vector field and its basic cohomology. Through a deformation-based approach, we show that a closed, orientable 3-manifold admits a quasi-Sasakian structure precisely when it is either Sasakian or arises as a Kähler mapping torus. In particular, every quasi-Sasakian structure in this setting can be deformed into a Sasakian or a co-Kähler one. This result leads to a complete classification of quasi-Sasakian manifolds in dimension three and highlights the geometric and topological features that distinguish the two cases.
We show that the parallel transport map over a reductive homogeneous space with natural torsion-free connection becomes an affine submersion with horizontal distribution. This generalizes one of the main results in the author's previous paper in the case of affine symmetric spaces. We also prove the compactness of the shape operators of the submanifold lifted by the parallel transport map. This improves a previous result by the author and generalizes some results of Terng-Thorbergsson and of Koike. Furthermore we propose two definitions for the regularized mean curvatures of affine Fredholm submanifolds in Hilbertable spaces and discuss their relations to the parallel transport map. In particular, each fiber of the parallel transport map over a reductive homogeneous space is shown to be minimal in both senses.
In this paper, we study the complex structures of complete hyperkähler four-manifolds of infinite topological type arising from the Gibbons-Hawking ansatz. We show that for almost all complex structures in the hyperkähler family, the manifold is biholomorphic to a hypersurface in ℂ^3 defined by an explicit entire function. For the remaining complex structures, we further prove that the manifold is biholomorphic to the minimal resolution of a singular surface in ℂ^3 under certain conditions. Thus, we partially extend LeBrun’s celebrated work [LeBrun, C.: Complete Ricci-flat Kähler metrics on ℂ ^n need not be flat. In Proc. Symp. Pure Math 52, 297–304 (1991)] to the context of countably many punctures.
We introduce the Courant algebroid lift, a new construction that takes a Courant algebroid together with a vector bundle connection and produces, when the connection is flat in the image of the anchor, a Courant algebroid. In general, this lift produces a Courant-like structure that we call a curved Courant algebroid. We start by establishing a hierarchy of Courant algebroid properties and their associated structures. In this setting, we introduce curved Courant algebroids, which we show to be related to connections with torsion and curved differential graded Lie algebras. We use this to provide a classification of exact curved Courant algebroids. We show that the Courant algebroid lift of an exact Courant algebroid yields a natural link between the Patterson-Walker metric and generalized geometry. By lifting non-exact Courant algebroids, we establish a relation of these lifts to Lie algebras, Poisson and special complex geometry. Finally, we show that Courant algebroid lifts provide a large class of examples of Courant algebroid actions.
We can show that the Kuranishi space of a pair (M,E) of a compact Kähler manifold M and its flat Hermitian vector bundle E is isomorphic to the direct product of the Kuranishi space of M and the Kuranishi space of E. We study non-Kähler case. We show that the Kuranishi space of a pair (M,E) of a complex parallelizable nilmanifold M and its trivial holomorphic vector bundle E is isomorphic to the direct product of the Kuranishi space of M and the Kuranishi space of E. We give examples of pairs (M,E) of nilmanifolds M with left-invariant abelian complex structures and their trivial holomorphic line bundles E such that the Kuranishi spaces of pairs (M,E) are not isomorphic to direct products of the Kuranishi spaces of M and the Kuranishi spaces of E.
This paper studies three-dimensional compact static manifolds with boundary and positive scalar curvature. We prove that, under a suitable bound on the Ricci curvature, the orientable quotient of the Nariai static manifold with boundary Nar_-1,1(𝕊^2) is the only such manifold with connected boundary, provided that the zero-level set of the potential is connected and does not intersect the boundary. We also establish a rigidity theorem for the upper hemisphere with the standard static potential, in the spirit of Cruz and Nunes.
We study the positive Hermitian curvature flow for left-invariant metrics on 2-step nilpotent Lie groups G with a left-invariant complex structure J. We describe the long-time behavior of the flow under the assumption that J[𝔤, 𝔤] is contained in the center of 𝔤 . We show that under our assumption the flow g_t exists for all positive t and (G,(1+t)^-1g_t) converges, in the Cheeger-Gromov topology, to a 2-step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups [21, 24]. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.
We define a Chern–Simons invariant of connections on stably trivial vector bundles over smooth manifolds, taking values in 3-forms modulo closed forms with integral cohomology class. We show an additivity property of this invariant for connections defined on a direct sum of bundles, under a certain block-diagonality condition on the curvature. As a corollary, we deduce an obstruction for conformally immersing a n-dimensional Riemannian manifold in a translation manifold of dimension n+1.
We classify Riemannian spin^c manifolds carrying a type I imaginary generalized Killing spinor, by explicitly constructing a parallel spinor on each leaf of the canonical foliation given by the Dirac current. We also provide a class of Riemannian spin^c manifolds carrying a type II imaginary generalized Killing spinor, by considering spacelike hypersurfaces of Lorentzian spin^c manifolds. We carry out much of the work in the setting of semi-Riemannian spin^c-manifolds carrying generalized Killing spinors, allowing us to draw conclusions in this setting as well. In this context, the Dirac current is not always a closed vector field. We circumvent this in even dimensions, by considering a modified Dirac current, which is closed in the cases when the original Dirac current is not. On the path to these results, we also study semi-Riemannian manifolds carrying closed and conformal vector fields.
We establish Willmore-type inequalities for bounded domains in complete non-compact Riemannian manifolds, under either asymptotic or integral Ricci curvature bounds. Those results recover a recent inequality of Jin-Yin [12, Theorem 1.3].
The desmic pencil of quartic surfaces is part of a beautiful, but mostly forgotten chapter of the classical theory of algebraic surfaces: it is the only non-degenerate pencil of quartic surfaces in ℙ^3 containing at least three completely reducible members. We observe in this note that it is closely related to the Weyl group of the root system F_4 , and can be recovered from a series of symmetric spaces deduced from the exceptional Lie algebras. We discuss the main properties of the pencil from this Lie theoretic point of view.
A ‘higher extremal Kähler metric’ is defined (motivated by analogy with the definition of an extremal Kähler metric) as a Kähler metric whose top Chern form equals a globally defined smooth function multiplied by its volume form such that the gradient of the smooth function is a holomorphic vector field. A special case of this kind of metric is a ‘higher constant scalar curvature Kähler (higher cscK) metric’ which is defined (again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric) as one whose top Chern form is a constant multiple of its volume form or equivalently whose top Chern form is harmonic. In our previous paper on higher extremal Kähler metrics we had looked at a certain class of minimal ruled surfaces called as ‘pseudo-Hirzebruch surfaces’ all of which contain two special divisors (viz. the zero and infinity divisors) and serve as the primary example manifolds in the momentum construction method (the Calabi ansatz procedure) which is used for producing explicit examples of the above-mentioned kinds of canonical Kähler metrics. We had proven that every Kähler class on such a surface admits a momentum-constructed higher extremal Kähler metric which is not higher cscK and we had further proven by using the ‘top Bando-Futaki invariant’ that higher cscK metrics do not exist in any Kähler class on the surface. In this paper we will see that if we allow our metrics to develop ‘conical singularities’ along at least one of the two special divisors of the surface then we do get ‘conical higher cscK metrics’ in each Kähler class of the surface by the momentum construction method. We will show that our constructed metrics satisfy the “polyhomogeneous condition” for conical Kähler metrics and we will interpret the conical higher cscK equation “globally on the surface” in terms of the currents of integration along the zero and infinity divisors. We will introduce the ‘top log Bando-Futaki invariant’ and then try to prove some standard expected results about it, with the final aim being to employ it to arrive at a certain linear relationship, that we conjecture must exist, between the cone angles of the conical singularities along the zero and infinity divisors.
With the notions of magnetic curvature and magnetic second fundamental form recently introduced by Assenza and Albers-Benedetti-Maier, respectively, we establish analogues of the Gauss, Ricci, and Codazzi-Mainardi compatibility equations from submanifold theory in the magnetic setting.
