Study gradient Ricci almost solitons as warped products.
problem Characterize gradient Ricci almost solitons as warped products.
method Use Bishop-O'Neill's formulas and Ricci-Hessian equations.
result Prove existence and rigidity results for gradient Ricci almost solitons.
The paper proves a new inequality for CR-warped product submanifolds in complex space forms.
problem Proving a new inequality for CR-warped product submanifolds in complex space forms.
method Developed a first Chen inequality for CR-warped product submanifolds in complex space forms.
result The bound is sharp and uniform in the sign of the holomorphic sectional curvature.
Generalizes O'Neill's equations to pseudo-Finsler submersions.
problem Extending Riemannian submersion equations to pseudo-Finsler geometry.
method Generalization of O'Neill's fundamental equations to pseudo-Finsler submersions and exploration of O'Neill tensors.
result Generalized fundamental equations for pseudo-Finsler submersions.
Paper unifies curvature concepts for Lie groupoids and algebroids.
problem Separate theories for Lie groups, bundles, and submersions.
method Using Lie groupoids with source-fibre metrics and Riemannian Lie algebroids.
result Derives sectional curvature formula for Lie groupoids and Lie algebroid curvature formulas.
This paper analyzes the Bochner formula for Riemannian flows and derives eigenvalue estimates.
problem Analyzing the Bochner formula for Riemannian flows and deriving eigenvalue estimates.
method The approach involves studying the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M, splitting it into two parts, and establishing eigenvalue estimates.
result Established an eigenvalue estimate of the basic Laplacian on basic forms, and discussed the limiting case of the estimate.
Paper derives a formula for curvature forms of submersions.
problem Understanding curvature forms of submersions.
method Using Karcher's formulation and shrinking fibers, derived a concise formula.
result As ε approaches 1, curvature forms approach sum of vertical and pullback curvature forms.
Given a finite dimensional manifold N, the group DiffS(N) of diffeomorphism of N which fall suitably rapidly to the identity, acts on the manifold B(M,N) of submanifolds on N of diffeomorphism type M where M is a compact manifold with dimM<dimN. For a right invariant weak …
Study of contact-complex Riemannian submersions in various manifold classes.
problem Characterizing submersions in specific manifold classes.
method Analyzing properties of O'Neill invariants and discussing integrability and minimality.
result Main properties of O'Neill invariants and integrability/minimality of fibres discussed.
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…
In this paper, we consider a Riemannian foliation whose normal bundle carries a parallel or harmonic basic form. We estimate the norm of the O'Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.
We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary mani…
We characterize compact eight-manifolds M which arise as internal spaces in N=1 flux compactifications of M-theory down to AdS3 using the theory of foliations, for the case when the internal part of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equiva…
Hyperkähler reduction explained via frame bundles.
problem Describing hyperkähler reduction using frame bundles.
method Using frame bundles and connections, tracing reductions and analyzing fibers.
result Fibers of hyperkähler reduction are totally geodesic.
If π:M→B is a Riemannian Submersion and M has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of…
We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor T vanishes on the invariant vertical distribution. We give n…
Study local equivalence of Riemannian submersions using differential invariants.
problem Local equivalence problem for Riemannian submersions under fiber-preserving isometries.
method Analysis of differential invariants for orbit submersions induced by a Killing field.
result Explicit formulas for A and H in terms of base data (gˉ,φ,Ω) and equivalence criterion. Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of…
In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation (M,F) with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac op…
The paper classifies isometric immersions between spheres of equal curvature.
problem Classifying isometric immersions between spheres of equal curvature.
method Defined coherent tangent bundles and classified wave fronts.
result Any isometric immersion of the n-sphere into the (n+1)-sphere of the same curvature is totally geodesic.
In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold…
In this paper, we give a new lower bound for the eigenvalues of the Dirac operator on a compact spin manifold. This estimate is motivated by the fact that in its limiting case a skew-symmetric tensor (see Equation \eqref{eq:16}) appears that can be identified geometrically with the O'Neill tensor of a Riemannian flow, …
Riemannian submersions can preserve positive intermediate Ricci curvature, but not necessarily.
problem Understanding the conditions under which Riemannian submersions preserve positive intermediate Ricci curvature.
method Analyzing the Gray--O'Neill Horizontal curvature equation and constructing perturbations of metrics.
result Riemannian submersions that do not preserve positive Ricci curvature are dense in the C1-topology. Study shows rigidity for spin^c manifolds with foliated boundaries.
problem Rigidity of Riemannian spin^c manifolds with foliated boundaries.
method Integral inequality based on geometric quantities.
result Characterization of equality cases for specific manifolds.
Study visibility properties of Kobayashi distance on unbounded domains.
problem Understanding visibility properties of Kobayashi distance on unbounded domains.
method Analyzing visibility properties in the context of Kobayashi hyperbolic domains, focusing on unbounded domains and their boundary behavior.
result Carathéodory-type extension theorem for biholomorphisms between planar domains, including infinitely-connected domains.
Warped product construction for Finsler manifolds with curvature conditions.
problem Constructing new Finslerian manifolds with specific curvature properties.
method Extending the warped product concept to Finsler metrics, particularly (α,β)-metrics. result Demonstrated the feasibility of constructing new Finslerian manifolds with prescribed curvature conditions.
Introduces Goldilocks domains with unique geometric properties.
problem Defines a new class of domains with specific growth conditions.
method Defines Goldilocks domains based on Kobayashi metric growth rates.
result Kobayashi metric exhibits visibility properties similar to negatively curved metrics.
The paper explores conformal submersions from Ricci solitons to Riemannian manifolds.
problem Characterizing conformal submersions from Ricci solitons.
method Analyzing properties of the O'Neill tensor, calculating Ricci tensors, and finding necessary conditions for harmonic maps.
result Necessary conditions for fibers and base manifold to be Ricci solitons, almost Ricci solitons, and Einstein.
The paper studies conjugate points on Lie groups with specific metrics.
problem Existence and properties of conjugate points on Lie groups with left-invariant metrics.
method Using reformulated index form in terms of adjoint action, the paper proves sufficient conditions for conjugate points and provides bounds and criteria.
result All geodesics in compact semisimple Lie groups have conjugate points, with upper and lower bounds on conjugate times.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Paper proves a fixed point formula and applies it to a new proof of Harish-Chandra's character formula.
problem Proving a fixed point formula for equivariant indices of elliptic differential operators.
method Fixed point formula for proper actions by connected semisimple Lie groups on manifolds.
result New proof of Harish-Chandra's character formula for discrete series representations.
Unified formula for surfaces in Euclidean or Lorentzian 3-space.
problem Describe surfaces in Euclidean or Lorentzian 3-space.
method Unified Kenmotsu-type formula for surfaces in Euclidean or Lorentzian 3-space.
result Unified single equation for Kenmotsu-type formulas in Euclidean and Lorentzian 3-space.
Two tropical gluing formulas help calculate Gromov-Witten invariants.
problem Calculating Gromov-Witten invariants of symplectic manifolds.
method Tropical geometry applied to exploded manifolds.
result Generalizes existing formulas for Gromov-Witten invariants.
Paper derives trace formula for magnetic Laplacian at zero energy.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
The Gauss formula is extended to various Laplacians on submanifolds.
problem Deriving formulas for Laplacians on submanifolds.
method Extending the Gauss formula to different types of Laplacians.
result Formulas for various Laplacians on submanifolds.
The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…
Note on new cancellation formulas for manifolds.
problem Generalizing anomaly cancellation formulas to manifolds.
method Proving new (a, b) type cancellation formulas and using transgression.
result Obtained characteristic forms with modularity properties.
The study identifies types of manifolds using variational formulas and integral-differential formulas.
problem Identifying specific types of manifolds based on curvature properties.
method Established variational formulas for Ricci curvature bounds and used them to identify manifolds.
result Constant curvature, Einstein, and Ricci parallel manifolds identified with specific formulas.
The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …
Formula calculates volume of two-bridge knots.
problem Calculating the volume of two-bridge knots.
method Derived from Hopf formula and Fox derivatives.
result Closed formula for the volume of two-bridge knots.
Formula connects surgeries to Seiberg-Witten invariants.
problem Understanding how surgeries affect Seiberg-Witten invariants.
method Proves surgery formulas for Seiberg-Witten invariants and families.
result Expresses new invariants in terms of original ones.
It has been shown that the Alvarez-Gaumeˊ-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…
Introduces a universal Bochner formula for scalar curvature.
problem None explicitly stated; focuses on a new formula.
method Introduces a universal Bochner formula.
result Contains special cases like stability inequality and Schrödinger-Lichnerowicz-type formula.
Extension formulae on almost complex manifolds studied with applications.
problem Understanding almost complex manifolds through extension formulae.
method Provided extension formulae and decompositions for almost complex manifolds.
result Studied (n,0)-forms, (n,0)-Dolbeault cohomology group, and (n,q)-forms. Formula calculates Gromov-Witten invariants for triple products.
problem Calculating Gromov-Witten invariants for complex geometrical structures.
method Developed a gluing formula for Gromov-Witten invariants in a triple product.
result Simplified gluing formula for easy explanation.
Calculates volumes of specific cone-manifolds using Schläfli formula.
problem Calculating volumes of specific cone-manifolds.
method Using Schläfli formula.
result Volume of the 732 link cone-manifolds and its cyclic coverings. Proves a formula for a special invariant of 4-manifolds.
problem Calculating the Bauer-Furuta invariant for connected sums of 4-manifolds.
method Uses a finite dimensional approximation of the Seiberg-Witten monopole map to derive a formula for the families Bauer-Furuta invariant of a fibrewise connected sum.
result Derives a general connected sum formula for the families Bauer-Furuta invariant.