In the mathematical field of Riemannian geometry,the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold,it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components,although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces,the scalar curvature is twice the Gaussian curvature,and completely characterizes the curvature of a surface. In higher dimensions,however,the scalar curvature only represents one particular part of the Riemann curvature tensor.
Shing-Tung Yau is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022,Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University.
Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow,who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem to the context of certain types of metric spaces.
In the mathematical field of differential geometry,there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds,although there has also been research into splitting of Lorentzian manifolds.
Branko Grünbaum was a Croatian-born mathematician of Jewish descent and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel.
Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
In mathematics,the Cartan–Dieudonnétheorem,named after Élie Cartan and Jean Dieudonné,establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described as the composition of at most n reflections.
In the mathematical field of metric geometry,Mikhael Gromov proved a fundamental compactness theorem for sequences of metric spaces. In the special case of Riemannian manifolds,the key assumption of his compactness theorem is automatically satisfied under an assumption on Ricci curvature. These theorems have been widely used in the fields of geometric group theory and Riemannian geometry.
In mathematics,the Reeb vector field,named after the French mathematician Georges Reeb,is a notion that appears in various domains of contact geometry including:
In mathematics,Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the m-dimensional sphere with its usual "round" metric. The theorem is named after the mathematician Marcel Berger,who derived it from an inequality proved by Jerry Kazdan.
Shoshichi Kobayashi was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds,transformation groups of geometric structures,and Lie algebras.
Ping Zhang is a mathematician specializing in graph theory. She is a professor of mathematics at Western Michigan University and the author of multiple textbooks on graph theory and mathematical proof.
Jeanne A. Nielsen Clelland is an American mathematician specializing in differential geometry and its applications to differential equations. She is a professor of mathematics at the University of Colorado Boulder,and the author of a textbook on moving frames,From Frenet to Cartan:The Method of Moving Frames.
Laura Christine Kinsey is an American mathematician specializing in topology. She is a professor of mathematics at Canisius College.
Introduction to the Theory of Error-Correcting Codes is a textbook on error-correcting codes,by Vera Pless. It was published in 1982 by John Wiley &Sons,with a second edition in 1989 and a third in 1998. The Basic Library List Committee of the Mathematical Association of America has rated the book as essential for inclusion in undergraduate mathematics libraries.
Helga Baum is a German mathematician. She is professor for differential geometry and global analysis in the Institute for Mathematics of the Humboldt University of Berlin.
Meike Maria Elisabeth Akveld is a Swiss mathematician and textbook author,whose professional interests include knot theory,symplectic geometry,and mathematics education. She is a tenured senior scientist and lecturer in the mathematics and teacher education group in the Department of Mathematics at ETH Zurich. She is also the organizer of the Mathematical Kangaroo competitions in Switzerland,and president of the Association Kangourou sans Frontières,a French-based international society devoted to the popularization of mathematics.
William Munger Boothby was an American mathematician and professor emeritus of mathematics at Washington University in St. Louis,known for his work in differential geometry including the book An introduction to differentiable manifolds and Riemannian geometry.
Dianne Carol Hansford is an American computer scientist known for her research on Coons patches in computer graphics and for her textbooks on computer-aided geometric design,linear algebra,and the mathematics behind scientific visualization. She is a lecturer at Arizona State University in the School of Computing and Augmented Intelligence,and the cofounder of a startup based on her research,3D Compression Technologies.
