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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

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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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The purpose of this book is to provide a self-contained account, accessible to the non-specialist, of algebra necessary for the solution of the integrability problem for transitive pseudogroup structures.Originally published in 1981.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Lie algebras. --- Ideals (Algebra) --- Pseudogroups. --- Global analysis (Mathematics) --- Lie groups --- Algebraic ideals --- Algebraic fields --- Rings (Algebra) --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Lie algebras --- Pseudogroups --- 512.81 --- 512.81 Lie groups --- Ideals (Algebra). --- Lie, Algèbres de. --- Idéaux (algèbre) --- Pseudogroupes (mathématiques) --- Ordered algebraic structures --- Analytical spaces --- Addition. --- Adjoint representation. --- Algebra homomorphism. --- Algebra over a field. --- Algebraic extension. --- Algebraic structure. --- Analytic function. --- Associative algebra. --- Automorphism. --- Bilinear form. --- Bilinear map. --- Cartesian product. --- Closed graph theorem. --- Codimension. --- Coefficient. --- Cohomology. --- Commutative ring. --- Commutator. --- Compact space. --- Complex conjugate. --- Complexification (Lie group). --- Complexification. --- Conjecture. --- Constant term. --- Continuous function. --- Contradiction. --- Corollary. --- Counterexample. --- Diagram (category theory). --- Differentiable manifold. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Discrete space. --- Donald C. Spencer. --- Dual basis. --- Embedding. --- Epimorphism. --- Existential quantification. --- Exterior (topology). --- Exterior algebra. --- Exterior derivative. --- Faithful representation. --- Formal power series. --- Graded Lie algebra. --- Ground field. --- Homeomorphism. --- Homomorphism. --- Hyperplane. --- I0. --- Indeterminate (variable). --- Infinitesimal transformation. --- Injective function. --- Integer. --- Integral domain. --- Invariant subspace. --- Invariant theory. --- Isotropy. --- Jacobi identity. --- Levi decomposition. --- Lie algebra. --- Linear algebra. --- Linear map. --- Linear subspace. --- Local diffeomorphism. --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monomorphism. --- Morphism. --- Natural transformation. --- Non-abelian. --- Partial differential equation. --- Pseudogroup. --- Pullback (category theory). --- Simple Lie group. --- Space form. --- Special case. --- Subalgebra. --- Submanifold. --- Subring. --- Summation. --- Symmetric algebra. --- Symplectic vector space. --- Telescoping series. --- Theorem. --- Topological algebra. --- Topological space. --- Topological vector space. --- Topology. --- Transitive relation. --- Triviality (mathematics). --- Unit vector. --- Universal enveloping algebra. --- Vector bundle. --- Vector field. --- Vector space. --- Weak topology. --- Lie, Algèbres de. --- Idéaux (algèbre) --- Pseudogroupes (mathématiques)

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This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Algebres de Clifford --- Clifford [Algebra's van ] --- Clifford algebras --- Fysica [Mathematische ] --- Fysica [Wiskundige ] --- Mathematische fysica --- Physics -- Mathematics --- Physics [Mathematical ] --- Physique -- Mathématiques --- Physique -- Méthodes mathématiques --- Wiskundige fysica --- Clifford, Algèbres de --- Spin, Nuclear --- Geometric algebras --- Clifford algebras. --- Spin geometry. --- Clifford, Algèbres de --- Spin geometry --- 514.76 --- Algebras, Linear --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Geometry of differentiable manifolds and of their submanifolds --- Global differential geometry --- Geometry --- Mathematical physics --- Topology --- Nuclear spin --- -Mathematics --- Géométrie --- Physique mathématique --- Spin nucléaire --- Topologie --- Mathematics --- Mathématiques --- Algebraic theory. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Binary icosahedral group. --- Binary octahedral group. --- Bundle metric. --- C*-algebra. --- Calabi conjecture. --- Calabi–Yau manifold. --- Cartesian product. --- Classification theorem. --- Clifford algebra. --- Cobordism. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Complete metric space. --- Complex manifold. --- Complex vector bundle. --- Complexification (Lie group). --- Covering space. --- Diffeomorphism. --- Differential topology. --- Dimension (vector space). --- Dimension. --- Dirac operator. --- Disk (mathematics). --- Dolbeault cohomology. --- Einstein field equations. --- Elliptic operator. --- Equivariant K-theory. --- Exterior algebra. --- Fiber bundle. --- Fixed-point theorem. --- Fourier inversion theorem. --- Fundamental group. --- Gauge theory. --- Geometry. --- Hilbert scheme. --- Holonomy. --- Homotopy sphere. --- Homotopy. --- Hyperbolic manifold. --- Induced homomorphism. --- Intersection form (4-manifold). --- Isomorphism class. --- J-invariant. --- K-theory. --- Kähler manifold. --- Laplace operator. --- Lie algebra. --- Lorentz covariance. --- Lorentz group. --- Manifold. --- Mathematical induction. --- Metric connection. --- Minkowski space. --- Module (mathematics). --- N-sphere. --- Operator (physics). --- Orthonormal basis. --- Principal bundle. --- Projective space. --- Pseudo-Riemannian manifold. --- Pseudo-differential operator. --- Quadratic form. --- Quaternion. --- Quaternionic projective space. --- Ricci curvature. --- Riemann curvature tensor. --- Riemannian geometry. --- Riemannian manifold. --- Ring homomorphism. --- Scalar curvature. --- Scalar multiplication. --- Sign (mathematics). --- Space form. --- Sphere theorem. --- Spin representation. --- Spin structure. --- Spinor bundle. --- Spinor field. --- Spinor. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Tangent bundle. --- Tangent space. --- Tensor calculus. --- Tensor product. --- Theorem. --- Topology. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Nuclear spin - - Mathematics --- -Clifford algebras.