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Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory.
128 Tensors: Geometry and Applications, J. M. Landsberg (2012, ISBN 978-0-8218-6907-9) 129 Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value Problems, Stuart P. Hastings, J. Bryce McLeod (2012, ISBN 978-0-8218-4694-0)
In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature. [1]
John "Jack" Marshall Lee (born September 2, 1950) is an American mathematician and professor at the University of Washington specializing in differential geometry. [ 1 ] Education
There is a close connection between 4 dimensional Conformal Geometry and 3 dimensional CR geometry associated with the study of CR manifolds.There is a naturally defined fourth order operator on CR manifolds introduced by C. Robin Graham and John Lee that has many properties similar to the classical Paneitz operator defined on 4 dimensional Riemannian manifolds. [1]
In differential geometry, one can attach to every point ... Lee, Jeffrey M. (2009), Manifolds and Differential Geometry, Graduate Studies in Mathematics, ...
Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics.
A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to real n-space R n. [2]A topological manifold is a locally Euclidean Hausdorff space.