Khovanov-Floer theories are shown to be invariant under mutation.
problem Invariance of Khovanov-Floer theories under Conway mutation.
method Spectral sequences from Khovanov homology and proofs of conjectures.
result Strong Khovanov-Floer theories are mutation-invariant.
In this paper we present several counterexamples to Rasmussen's conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from Ozsv{á}th-Szab{ó} Floer homology. The counterexamples are twisted Whitehead doubles of the (2,2n+1) torus knots.
New deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
We study contact structures compatible with genus one open book decompositions with one boundary component. Any monodromy for such an open book can be written as a product of Dehn twists around dual non-separating curves in the once-punctured torus. Given such a product, we supply an algorithm to determine whether the …
New knot concordance invariants derived from regions in the plane.
problem Knot concordance and distinguishing knots from thin or algebraic ones.
method Associate invariants to regions in the plane, compute for specific knots, and use to obstruct concordances.
result Compute and use new invariants to obstruct concordances to specific types of knots.
In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot K. We present a formula for the filtered chain homotopy type of HFK(D(+,K,t)) in terms of the invariants for K, where D(+,K,t) denotes the t-twisted positive-clasped Whitehead double of K. In pa…
The thesis examines when double branched covers of alternating knots arise via Dehn surgery.
problem When the double branched cover of an alternating knot can arise by Dehn surgery on a knot in S3. method Surgery obstruction combining Donaldson's Diagonalization Theorem and Heegaard Floer homology.
result Alternating knots with an unknotting crossing have unknotting number one.
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
Study finds six homogeneous surfaces with multiple invariant connections.
problem Characterizing homogeneous surfaces with invariant connections.
method Computed all simply connected homogeneous and infinitesimally homogeneous surfaces.
result Found six non-equivalent surfaces with multiple invariant connections.
A manifold's canonical involution defines a projection of connections.
problem Defining a projection of connections on (J2=±1)-metric manifolds. method Introducing a canonical involution to project connections.
result The projection sends Levi Civita to first canonical connection.
Study non-integrable distributions with various affine connections.
problem Characterize non-integrable distributions in Riemannian manifolds with different connections.
method Obtain Gauss, Codazzi, and Ricci equations for non-integrable distributions with semi-symmetric metric, non-metric, and statistical connections.
result Find new examples of Einstein and distributions with constant scalar curvature.
Study various connections on (J2=±1)-metric manifolds.
problem Characterize and compare linear connections on (J2=±1)-metric manifolds. method Examined first canonical, Chern, well adapted, Levi Civita, Kobayashi-Nomizu, Yano, Bismut, and totally skew-symmetric torsion connections.
result Every connection studied is a canonical connection when it exists and is adapted.
Paper explores connection cochain in abelian extensions and its relation to connection forms.
problem Understanding the connection cochain in abelian extensions.
method Apply Moriyoshi's connection cochain concept to abelian extensions and relate it to connection 1-forms.
result Established the relationship between connection cochain and connection 1-forms in abelian extensions.
Explores connective spaces, their representations, foliations, and relations to diffeological spaces.
problem Developing a comprehensive theory of connective spaces and their properties.
method Historical context, development of connective representation and foliation, generalization of connectivity order, study of functorial relations with diffeological spaces.
result Connectivity order generalized to all connectivity spaces and connective foliations.
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying algebraic Ricci solitons on three-dimensional Lorentzian Lie groups.
method Computed canonical and Kobayashi-Nomizu connections and their curvatures; defined algebraic Ricci solitons.
result Classified algebraic Ricci solitons on specific Lorentzian Lie groups.
New normalization condition for sub-Riemannian connections.
problem Normalizing connections on sub-Riemannian manifolds.
method Formulated in terms of Cartan connections, depends on curvature's first degree of homogeneity.
result A compatible partial affine connection can be uniquely extended to a full affine connection and a grading of the tangent bundle.
Paper proves unique Finsler connections for scalar forms.
problem Existence and uniqueness of Finsler connections.
method Pullback approach to global Finsler geometry, study of horizontally recurrent connections.
result Existence and uniqueness of horizontally recurrent Finsler connections for scalar forms.
Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
A new connection defined in sub-Riemannian geometry.
problem Defining a canonical connection in sub-Riemannian contact geometry.
method Inspired by Levi-Civita connection, constructs an affine connection.
result Compares with Tanaka-Webster connection in 3D.
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
The paper proves monotonicity formulas for minimal connections and their applications.
problem Understanding critical points of volume functionals in Riemannian geometry.
method Developed monotonicity formulas for minimal connections under specific conditions.
result Established vanishing theorems for minimal connections on Euclidean spaces and dDT connections on G2-manifolds.
Knots connected via a trivial band sum to connected sum.
problem Conditions for band-connected sum to equal connected sum.
method Analyzing knots and bands to determine conditions for equality.
result A band is trivial if and only if a band-connected sum equals a connected sum.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
Study of multiplicative connections in Lie groupoids.
problem Defining and understanding multiplicative connections in Lie groupoids.
method Definition and study of multiplicative connections satisfying compatibility with the groupoid structure.
result Identification of the obstruction to the existence of a multiplicative connection.
Study on submanifolds in generalized Sasakian-space-forms with various connections.
problem Analyzing submanifolds in generalized Sasakian-space-forms with different connections.
method Examines submanifolds in generalized Sasakian-space-forms with semisymmetric metric, non-metric, Schouten-van Kampen, and Tanaka-webster connections.
result Provides results on submanifolds in generalized Sasakian-space-forms with respect to various connections.
Develops torsion dual connections for statistical manifolds.
problem Defining statistical manifolds using dual connections.
method Introduces torsion dual connections and proves their properties.
result Curvature tensor of torsion dual connections has specific divergence.
Paper studies well adapted connections for specific metric manifolds.
problem Characterizing well adapted connections for (J2=±1)-metric manifolds. method Proves existence and derives explicit formula for well adapted connections in four geometries.
result Characterizes coincidence of well adapted connections with Levi Civita and Chern connections.
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
Automorphisms of symplectic connections are studied, with examples showing not all are symplectic.
problem Characterizing infinitesimal automorphisms of symplectic connections.
method Analyzing conditions for infinitesimal automorphisms to be symplectic vector fields.
result Examples show that not all infinitesimal automorphisms are symplectic.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Extends dual connections to vector bundles, interpreting Amari tensor geometrically.
problem No specific problem stated; extends dual connections concept.
method Uses Cartan decompositions of Lie algebras to generalize dual connections to vector bundles.
result Geometric interpretation of Amari tensor as a connection form term generating dilations.
Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
Defines a new natural connection on Riemannian Π-manifolds.
problem Characterizing natural connections on Riemannian Π-manifolds.
method Introducing and analyzing the first natural connection with torsion.
result Relations between the first natural connection and Levi-Civita connection are established.
This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.
problem Determining the global meromorphic connection based on specified local behavior at singular points.
method Expository discussion of various problems related to meromorphic connections with specified local behavior, including Deligne-Simpson and rigidity problems.
result The existence and nonemptiness of moduli spaces of meromorphic connections with specified local behavior.
New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the g…
We assume a vector bundle p:E→M with a general linear connection K and a classical linear connection $\Lam$ on M. We prove that all classical linear connections on the total space E naturally given by $(\Lam, K)$ form a 15-parameter family. Further we prove that all connections on J1E naturally given by…
Paper develops a unified framework for Lie algebroid connections on various bundles.
problem Unified framework for Lie algebroid connections on vector and principal bundles.
method Generalized Atiyah algebroid structure and its short exact sequence.
result Explicit constructions of Atiyah classes for Lie algebroid connections.
Characterizes connections on multivariate normal distributions.
problem Characterizing connections on statistical manifold of multivariate normal distributions.
method Analyzes statistical manifold (N,gF,ablaA,ablaA∗) of multivariate normal distributions. result The Amari-Chentsov connection ablaA is characterized by conjugate symmetry. Classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on three-dimensional Lorentzian Lie groups.
method Examined canonical and perturbed canonical connections, as well as Kobayashi-Nomizu connections and perturbed Kobayashi-Nomizu connections.
result Classified affine Ricci solitons on three-dimensional Lorentzian Lie groups with product structure.