In this paper, we investigate Chen’s δ-invariant for partially slant (PS) submanifolds of complex space forms. A PS-submanifold admits an orthogonal decomposition of the tangent bundle into a proper slant distribution and an arbitrary ambiguous distribution. Using the Gauss equation together with algebraic optimization techniques, we derive a Chen-type inequality relating the δ-invariant to the squared mean curvature, the holomorphic sectional curvature of the ambient space, and the slant angle of the slant distribution. Unlike the classical Chen inequality for slant submanifolds, the obtained estimate contains an additional term reflecting the contribution of the ambiguous distribution. Several corollaries are derived, including dimension-dependent bounds and special cases corresponding to hemi-slant and semi-slant submanifolds. The equality case is also characterized in terms of the structure of the shape operators. These results provide a natural extension of Chen-type inequalities to the broader framework of partially slant geometry in Kähler manifolds.

Abstract Logarithmic and ‐tangent bundles provide a versatile framework for addressing singularities in geometry. Introduced by Deligne and Melrose, these modified bundles resolve singularities by reframing singular vector fields as well‐behaved sections of these singular bundles . This approach has gained significant attention in symplectic geometry, particularly through its applications to the study of Poisson manifolds that are symplectic away from a hypersurface (‐symplectic forms). In this paper, we investigate the conditions under which these singular tangent bundles are isomorphic to the tangent bundle or other singular bundles, analyzing in detail the low‐dimensional case and the case of spheres. We also examine the existence of geometric structures in light of these conditions. Furthermore, we establish a Poincaré–Hopf theorem for the ‐tangent bundle, offering new insights into the interplay between singular structures and topological invariants.

We begin by studying the Riemannian geometry of the tangent Lie group $TG$ associated with a Lie group $G$ whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on $G$. We establish the relationship between the sectional curvatures of $G$ and those of $TG$. Next, we define a Randers metric on $G$ from a left-invariant Riemannian metric and a left-invariant vector field, and lift it vertically and completely to $TG$. We investigate the conditions under which this Randers metric is of Berwald and Douglas type, respectively, and compute the flag curvatures in the Berwald case. In an addendum, we discuss geodesic vectors and bi-invariant Riemannian metrics on these Lie groups, highlighting the special unimodularity conditions. Finally, we provide explicit formulas for the Riemannian curvature tensor on the tangent bundle of such a Lie group.

The problem studied in this paper is related to the f-harmonicity with potential of a vector field from a Riemannian to its tangent bundle TM equipped with the Sasaki metric. We show that an f-harmonic vector field with potential is parallel if and only if f is constant function. We also characterize the f-harmonic vector fields with potential on some three Riemannian Lie groups.

In this paper, we investigate the gradient Schouten soliton structure on the tangent bundle $TM$ of a Riemannian manifold endowed with the Sasaki metric ${}^{S}g$. The study is carried out with respect to an adapted frame. We derive necessary and sufficient conditions for the quadruples $(TM, {}^{S}g, {}^{V}f, \lambda)$ and $(TM, {}^{S}g, {}^{C}f, \lambda)$ to define gradient Schouten solitons.

Algebraists asked whether or not an operator on the module of smooth sections of the tangent bundle over the commutative ring of smooth functions of a smooth (orientable) manifold (can be any piece of a compact or a complete manifold) can be characterized by two axioms. In this note we confirm this for any smooth manifold M M under the assumption that H 1 ( M , R ) = { 0 } H^1(M, \mathbb {R})=\{0\} .

Abstract We provide an alternative construction of the quasi‐Fuchsian flows introduced by Ghys. Our approach is based on the coupled vortex equations that allows to see these flows as thermostats on the unit tangent bundle of the Blaschke metric, uniquely determined by a conformal class and a holomorphic quadratic differential. We also give formulas for the marked length spectrum of a quasi‐Fuchsian flow in the thermostat parametrization.

Abstract We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgery was first used by Smale [Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73 (6) (1967), 747–817] and Mañé [Contributions to the stability conjecture. Topology 17 (4) (1978), 383–396] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies, and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a two-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.

Cartan geometry provides a unifying algebraic construction of curvature and torsion, based on an underlying model Lie algebra; a viewpoint that can be extended naturally to the higher algebraic structures underlying supergravity. We present a Cartan-geometric framework for generalized geometries governed by a differential graded Lie algebra, extending previous results. The extended tangent bundle admits the action of both a global duality group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">G</a:mi> </a:math> and a local gauge group <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>H</d:mi> </d:math> . This algebraic structure is implemented via a brane current algebra; the phase space Poisson structure of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>p</f:mi> </f:math> -branes. Within this Cartan-inspired framework, we define a hierarchy of generalized connections and compute their linearized torsion and curvature tensors, including the higher curvatures required by the tensor hierarchy. This provides a systematic construction of curvature and torsion tensors in generic generalized geometries.

Abstract Developments in Carrollian gravity and holography necessitate the use of singular Carroll vector fields, a feature that cannot be accommodated within standard Carrollian geometry. We introduce Carrollian Lie algebroids as a framework to study such singular Carrollian geometries. In this approach, we define the Carroll distribution as the image of the kernel of the degenerate metric under the anchor map, i.e. the map from the Lie algebroid to the tangent bundle of the manifold. The Carroll distribution is, in general, a singular Stefan–Sussmann distribution that will fluctuate between rank-1 and rank-0, and so captures the notion of a singular Carroll vector field. As an example, we show that an invariant Carrollian structure on a principal bundle leads to a Carrollian structure on the associated Atiyah algebroid that will, in general, have a singular Carroll distribution. Mixed null-spacelike hypersurfaces, under some simplifying assumptions, also lead to examples of Carrollian Lie algebroids. Furthermore, we establish the existence of compatible connections on Carrollian Lie algebroids, and as a direct consequence, we conclude that Carrollian manifolds can always be equipped with compatible affine connections.

We show that the normal points of a cubic hypersurface in projective space have canonical singularities unless the hypersurface is an iterated cone over an elliptic curve. As an application, we give a simple linear algebraic description of all the hyperplane sections of projectivized tangent bundle of projective space, hence describing hyperplane sections of a rational homogeneous manifold of Picard rank 2 2 . This also simplifies and extends recent results of Mazouni-Nagaraj in higher dimensions. We also compute the Chow ring of these hyperplane sections.

The purpose of this note is twofold. First we show that, for weakly differentiable maps between Riemannian manifolds of any dimension, a smallness condition on a Morrey-norm of the gradient is sufficient to guarantee that the pulled-back tangent bundle is trivialised by a finite-energy frame over simply connected regions in the domain. This is achieved via new structure equations for a connection introduced by Rivière in the study of weakly harmonic maps, combined with Coulomb-frame methods and the Hardy-BMO duality of Fefferman-Stein. We also prove that for weakly harmonic maps from domains of any dimension into closed homogeneous targets, a smallness condition on the BMO seminorm of the map is sufficient to obtain full regularity.

We analyse the {alpha ^{prime },}^2 corrections to the supersymmetry algebra constructed by Bergshoeff–de Roo for heterotic compactifications on textrm{SU}(3) manifolds. The geometry is complex and conformally balanced. Starting from these supersymmetry constraints, we derive the equations of motions and find that the graviton equation contains an extraneous term which can be set to zero after gauge fixing. The curvature of the tangent bundle connection acquires a nonzero (0, 2) component and so does not satisfy the instanton equation, showing that the tangent bundle instanton condition does not persist beyond first order in {alpha ^{prime },}.

In this study, we introduce the notion of an m ‐modified conformal vector field, which represents a natural generalization of the classical conformal vector field concept. This extension allows for a broader class of vector fields that conform to modified transformation rules under conformal changes in geometry. Specifically, we explore the necessary and sufficient conditions under which the vector fields V V , C V , C V + γ F , and V u (also known as the Liouville vector field) qualify as m ‐modified conformal vector fields within the framework of the synectic lift metric on the tangent bundle. The investigation into these conditions requires careful analysis using geometric tools and algebraic methods. In our calculations, we employ an adapted frame, which significantly streamlines the computations and provides multiple advantages for the geometric interpretations. Through this approach, we are able to derive deeper insights into the structure of these vector fields and their role within the broader context of differential geometry.

Plasma kinematics is typically performed on either the mass shell or a lab bundle, seven-dimensional phase spaces equipped with a Vlasov vector field. This choice of phase space encodes the parametrisation of the differential equations governing particle dynamics. By replacing the Vlasov vector field and related quantities over a phase space with a bivector on an eight-dimensional conic sub-bundle of the tangent bundle, we construct a parameter-free version of Vlasov theory. The advantages of this formalism include compatibility with light-like and ultra-relativistic particles, non-metric connections, and metric-free or pre-metric theories. Additionally, this formalism applies to theories where no time phase space can exist for topological reasons, e.g. when we wish to consider all geodesics, including space-like geodesics. We also use this formalism to derive a simple formula which enables the transformation of a Vlasov field from one phase space to another in such a way that the trajectories of the particles in the base space are unchanged. Our formalism also has implications for numerical simulations. By extending quantities such as the particle density form onto an eight-dimensional conic sub-bundle, we can extend existing numerical integrators, such as the agile numerical integrator, to relativistic scenarios. The extension of the particle density form also allows for the generalisation of the current 3-form onto the conic bundle. We also discuss the complicated relationship between the stress-energy form and phase space, how our formalism illustrates this, and the corresponding challenges in developing a parameter-free Einstein-Vlasov system. The relationship between our formalism and sprays or semi-sprays is explored, and examples from Finsler geometry are given.

Determining the reachable set for a given nonlinear system is critically important for autonomous trajectory planning for reach-avoid applications and safety critical scenarios. Providing the reachable set is generally impossible when the dynamics are unknown, so we calculate underapproximations of such sets using local dynamics at a single point and bounds on the rate of change of the dynamics determined from known physical laws. Motivated by scenarios where an adverse event causes an abrupt change in the dynamics, we attempt to determine a provably reachable set of states without knowledge of the dynamics. This paper considers systems which are known to operate on a manifold. Underapproximations are calculated by utilizing the aforementioned knowledge to derive a guaranteed set of velocities on the tangent bundle of a complete Riemannian manifold that can be reached within a finite time horizon. We then interpret the underapproximated set as a control system; the trajectories of this control system provide us with a guaranteed set of reachable states the unknown system can reach within a given time. The results are general enough to apply on systems that operate on any complete Riemannian manifold. To illustrate the practical implementation of our results, we apply our algorithm to a three-dimensional rotational system which lives on the abstract set of special orthogonal matrices.

Abstract A causal manifold $(M,\gamma )$ is a manifold $M$ endowed with a closed convex proper cone $\gamma $ in the tangent bundle $TM$ satisfying suitable properties. Let $\lambda ={\gamma }^{\circ a}\subset T^{*}M$ be the antipodal of the polar cone of $\gamma $. An open set $U$ of $M$ is called $\gamma $-open if its Whitney normal cone contains $\textrm{Int}(\gamma )$. Similarly, $U$ is called $\lambda $-open if the micro-support of the constant sheaf on $U$ is contained in $\lambda $. We begin by proving that the two notions coincide. Next, we prove that if $(M,\gamma )$ admits a “future time function” (in particular if it is globally hyperbolic) the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $\lambda $ and the derived category of sheaves on $M_\gamma $, the manifold $M$ endowed with the $\gamma $-topology. This generalizes a result of [5], which treated the case of a constant cone in a vector space.

Let $TM$ be the tangent bundle over an $F$-Kählerian manifold endowed with Berger type deformed Sasaki metric $g_{BS}$. In this paper, we obtain the Levi-Civita connection of this metric and study geodesics on the tangent bundle $TM$ and $F$-unit tangent bundle $T_{1,F}M.$ Secondly, we characterize the geodesic curvatures on $T_{1,F}M$. Finally, we present some conditions for a vector field $\xi :M\rightarrow TM$ to be harmonic and study the harmonicity of the canonical projection $\pi :TM\rightarrow M$. In addition, we search the harmonicity of the Berger type deformed Sasaki metric $g_{BS}$ and the Sasaki metric $g_{S}$ with respect to each other.

In this work, we study inequalities and enumerative formulas for flags of Pfaff Systems on [Formula: see text]. More specifically, we establish a bound for the number of independent twisted [Formula: see text]-forms that leave invariant a one-dimensional holomorphic foliation and deduce inequalities that relate the degrees in the flags, which can be interpreted as a version of the Poincaré problem for flags. Moreover, by restricting to a flag of specific holomorphic foliations/distributions, we obtain inequalities involving the degrees. As a consequence, we obtain stability results for the tangent sheaf of some rank two holomorphic foliations/ distributions.

Abstract We show that the Lipschitz-Free Space over a connected orientable n -dimensional Riemannian manifold M is isometrically isomorphic to a quotient of $$L^1(M,TM)$$ L 1 ( M , T M ) , the integrable sections of the tangent bundle TM , if M is either complete or lies isometrically inside a complete manifold N . Two functions are deemed equivalent in this quotient space if their difference has distributional divergence zero. This quotient is the pre-annihilator of the exact essentially bounded currents, and if M is simply connected, one may replace “exact” with “closed” currents.