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In this volume, which is dedicated to H. Seifert, are papers based on talks given at the Isle of Thorns conference on low dimensional topology held in 1982.

Low-dimensional topology --- 515.162 --- Topology, Low-dimensional --- Algebraic topology --- Manifolds (Mathematics) --- 515.162 Low-dimensional manifold topology. Topological surfaces. Topological 3-manifolds, 4-manifolds. Knots. Links. Braids --- Low-dimensional manifold topology. Topological surfaces. Topological 3-manifolds, 4-manifolds. Knots. Links. Braids --- Congresses --- Differential topology --- Congresses. --- Low-dimensional topology - Congresses

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The present book contains 14 papers published in the Special Issue “Differential Geometry” of the journal Mathematics. They represent a selection of the 30 submissions. This book covers a variety of both classical and modern topics in differential geometry. We mention properties of both rectifying and affine curves, the geometry of hypersurfaces, angles in Minkowski planes, Euclidean submanifolds, differential operators and harmonic forms on Riemannian manifolds, complex manifolds, contact manifolds (in particular, Sasakian and trans-Sasakian manifolds), curvature invariants, and statistical manifolds and their submanifolds (in particular, Hessian manifolds). We wish to mention that among the authors, there are both well-known geometers and young researchers. The authors are from countries with a tradition in differential geometry: Belgium, China, Greece, Japan, Korea, Poland, Romania, Spain, Turkey, and United States of America. Many of these papers were already cited by other researchers in their articles. This book is useful for specialists in differential geometry, operator theory, physics, and information geometry as well as graduate students in mathematics.

statistical structure --- constant ratio submanifolds --- Euclidean submanifold --- framed helices --- Sasakian statistical manifold --- L2-harmonic forms --- Hodge–Laplacian --- complete connection --- concircular vector field --- cylindrical hypersurface --- k-th generalized Tanaka–Webster connection --- Casorati curvature --- symplectic curves --- generalized 1-type Gauss map --- rectifying submanifold --- manifold with singularity --- ruled surface --- Minkowski plane --- compact complex surfaces --- conjugate connection --- T-submanifolds --- L2-Stokes theorem --- inextensible flow --- shape operator --- generalized normalized ?-Casorati curvature --- Sasakian manifold --- centrodes --- circular helices --- non-flat complex space form --- invariant --- Frenet frame --- Darboux frame --- trans-Sasakian 3-manifold --- singular points --- symplectic curvatures --- Kähler–Einstein metrics --- conjugate symmetric statistical structure --- sectional ?-curvature --- circular rectifying curves --- developable surface --- capacity --- Ricci soliton --- Reeb flow symmetry --- Minkowskian pseudo-angle --- conical surface --- lie derivative --- position vector field --- pinching of the curvatures --- Hessian manifolds --- Minkowskian angle --- Hessian sectional curvature --- Minkowskian length --- lightlike surface --- affine sphere --- concurrent vector field --- slant --- affine hypersurface --- anti-invariant --- statistical manifolds --- Ricci operator --- C-Bochner tensor --- Ricci curvature --- real hypersurface --- scalar curvature --- framed rectifying curves

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This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

Topology --- Differential geometry. Global analysis --- Geometry, Hyperbolic --- Three-manifolds (Topology) --- Géométrie hyperbolique --- Variétés topologiques à 3 dimensions --- Geometry, Hyperbolic. --- 514.1 --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Non-Euclidean --- General geometry --- Three-manifolds (Topology). --- 514.1 General geometry --- Géométrie hyperbolique --- Variétés topologiques à 3 dimensions --- 3-sphere. --- Abelian group. --- Affine space. --- Affine transformation. --- Atlas (topology). --- Automorphism. --- Basis (linear algebra). --- Bounded set (topological vector space). --- Brouwer fixed-point theorem. --- Cartesian coordinate system. --- Characterization (mathematics). --- Compactification (mathematics). --- Conformal map. --- Contact geometry. --- Curvature. --- Cut locus (Riemannian manifold). --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Disk (mathematics). --- Divisor (algebraic geometry). --- Dodecahedron. --- Eigenvalues and eigenvectors. --- Embedding. --- Euclidean space. --- Euler number. --- Exterior (topology). --- Facet (geometry). --- Fiber bundle. --- Foliation. --- Fundamental group. --- Gaussian curvature. --- Geometry. --- Group homomorphism. --- Half-space (geometry). --- Holonomy. --- Homeomorphism. --- Homotopy. --- Horocycle. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hyperboloid model. --- Interior (topology). --- Intersection (set theory). --- Isometry group. --- Isometry. --- Jordan curve theorem. --- Lefschetz fixed-point theorem. --- Lie algebra. --- Lie group. --- Line (geometry). --- Linear map. --- Linearization. --- Manifold. --- Mathematical induction. --- Metric space. --- Moduli space. --- Möbius transformation. --- Norm (mathematics). --- Pair of pants (mathematics). --- Piecewise linear manifold. --- Piecewise linear. --- Poincaré disk model. --- Polyhedron. --- Projection (linear algebra). --- Projection (mathematics). --- Pseudogroup. --- Pullback (category theory). --- Quasi-isometry. --- Quotient space (topology). --- Riemann mapping theorem. --- Riemann surface. --- Riemannian manifold. --- Sheaf (mathematics). --- Sign (mathematics). --- Simplicial complex. --- Simply connected space. --- Special linear group. --- Stokes' theorem. --- Subgroup. --- Subset. --- Tangent space. --- Tangent vector. --- Tetrahedron. --- Theorem. --- Three-dimensional space (mathematics). --- Topological group. --- Topological manifold. --- Topological space. --- Topology. --- Transversal (geometry). --- Two-dimensional space. --- Uniformization theorem. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Topologie algébrique --- Topologie combinatoire --- Algebraic topology. --- Combinatorial topology. --- Variétés topologiques --- Geometrie --- Theorie des noeuds

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Since Poincaré's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area.The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.

Differential geometry. Global analysis --- Manifolds (Mathematics) --- Piecewise linear topology --- Triangulating manifolds --- Variétés (Mathématiques) --- Topologie linéaire par morceaux --- 515.16 --- Manifolds, Triangulating --- PL topology --- Topology --- Geometry, Differential --- Topology of manifolds --- Piecewise linear topology. --- Triangulating manifolds. --- Manifolds (Mathematics). --- 515.16 Topology of manifolds --- Variétés (Mathématiques) --- Topologie linéaire par morceaux --- Triangulation. --- Triangulation --- Affine space. --- Algebraic topology (object). --- Approximation. --- Associative property. --- Automorphism. --- Big O notation. --- CW complex. --- Calculation. --- Cap product. --- Cartesian product. --- Category of sets. --- Chain complex. --- Classification theorem. --- Classifying space. --- Cobordism. --- Codimension. --- Cofibration. --- Cohomology. --- Connected space. --- Continuous function (set theory). --- Continuous function. --- Counterexample. --- Diffeomorphism. --- Differentiable manifold. --- Differential structure. --- Differential topology. --- Dimension (vector space). --- Direct proof. --- Disjoint union. --- Elementary proof. --- Embedding. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Fiber bundle. --- Fibration. --- General position. --- Geometry. --- Group homomorphism. --- H-cobordism. --- H-space. --- Handle decomposition. --- Handlebody. --- Hauptvermutung. --- Hausdorff space. --- Hilbert cube. --- Homeomorphism group. --- Homeomorphism. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Inclusion map. --- Injective function. --- Invertible matrix. --- K-cell (mathematics). --- Kan extension. --- Linear subspace. --- Linear topology. --- Manifold. --- Mapping cylinder. --- Mathematical induction. --- Mathematician. --- Metric space. --- Morse theory. --- Neighbourhood (mathematics). --- Open set. --- Partition of unity. --- Piecewise linear manifold. --- Piecewise linear. --- Poincaré conjecture. --- Polyhedron. --- Principal bundle. --- Product metric. --- Pushout (category theory). --- Regular homotopy. --- Retract. --- Sheaf (mathematics). --- Simplicial complex. --- Smoothing. --- Spin structure. --- Stability theory. --- Stable manifold. --- Standard map. --- Submanifold. --- Submersion (mathematics). --- Subset. --- Surgery exact sequence. --- Surjective function. --- Theorem. --- Topological group. --- Topological manifold. --- Topological space. --- Topology. --- Transversal (geometry). --- Transversality (mathematics). --- Transversality theorem. --- Union (set theory). --- Uniqueness theorem. --- Vector bundle. --- Zorn's lemma. --- Variétés topologiques

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This book presents the recent developments in the field of geometric inequalities and their applications. The volume covers a vast range of topics, such as complex geometry, contact geometry, statistical manifolds, Riemannian submanifolds, optimization theory, topology of manifolds, log-concave functions, Obata differential equation, Chen invariants, Einstein spaces, warped products, solitons, isoperimetric problem, Erdös–Mordell inequality, Barrow’s inequality, Simpson inequality, Chen inequalities, and q-integral inequalities. By exposing new concepts, techniques and ideas, this book will certainly stimulate further research in the field.

Research & information: general --- Mathematics & science --- Erdös–Mordell inequality --- Barrow’s inequality --- triangle --- interior point --- q-integral inequality --- strongly preinvex function --- Simpson inequality --- quantum integral --- statistical warped product submanifold --- statistical manifold --- B.Y.Chen inequality --- Casorati curvatures --- statistical soliton --- Wintgen inequality --- generalized complex space form --- generalized Sasakian space form --- Lagrangian submanifold --- Legendrian submanifold --- Einstein manifold --- sectional curvature --- Betti number --- Tachibana number --- spherical space form --- slant submanifolds --- generalized Sasakian space forms --- closed form --- conformal form --- Maslov form --- legendrian submanifolds --- sasakian space forms --- obata differential equation --- isometric immersion --- Casorati curvature --- statistical submanifold --- holomorphic statistical manifold --- clifford minimal hypersurfaces --- sasakian structure --- integral inequalities --- reeb function --- contact vector field --- δ(2,2)-invariant --- Chen inequalities --- Lagrangian submanifolds --- quaternionic space forms --- complex space forms --- warped product --- sphere theorem --- Laplacian --- inequalities --- diffeomorphic --- Ricci curvature --- skew CR-warped product submanifolds --- complex space form --- CR-warped product submanifolds --- semi slant warped product submanifolds --- almost Hermitian manifolds --- Kähler identities --- Lefschetz operator --- isoperimetric problem --- minimal perimeter --- log-concave functions --- isotropic measure --- extrinsic principal tangential directions --- principal first normal directions --- Erdös–Mordell inequality --- Barrow’s inequality --- triangle --- interior point --- q-integral inequality --- strongly preinvex function --- Simpson inequality --- quantum integral --- statistical warped product submanifold --- statistical manifold --- B.Y.Chen inequality --- Casorati curvatures --- statistical soliton --- Wintgen inequality --- generalized complex space form --- generalized Sasakian space form --- Lagrangian submanifold --- Legendrian submanifold --- Einstein manifold --- sectional curvature --- Betti number --- Tachibana number --- spherical space form --- slant submanifolds --- generalized Sasakian space forms --- closed form --- conformal form --- Maslov form --- legendrian submanifolds --- sasakian space forms --- obata differential equation --- isometric immersion --- Casorati curvature --- statistical submanifold --- holomorphic statistical manifold --- clifford minimal hypersurfaces --- sasakian structure --- integral inequalities --- reeb function --- contact vector field --- δ(2,2)-invariant --- Chen inequalities --- Lagrangian submanifolds --- quaternionic space forms --- complex space forms --- warped product --- sphere theorem --- Laplacian --- inequalities --- diffeomorphic --- Ricci curvature --- skew CR-warped product submanifolds --- complex space form --- CR-warped product submanifolds --- semi slant warped product submanifolds --- almost Hermitian manifolds --- Kähler identities --- Lefschetz operator --- isoperimetric problem --- minimal perimeter --- log-concave functions --- isotropic measure --- extrinsic principal tangential directions --- principal first normal directions