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Nash-Kuiper Theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (''C''1) embeddings and the second for embeddings that are analytic or smooth of class ''Ck'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The ''C''1 theorem was published in 1954, the ''Ck''-theorem in 1956. The real analytic theorem was first treated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Forbes Nash Jr
John Forbes Nash Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory, real algebraic geometry, differential geometry, and partial differential equations. Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (popularly known as the Nobel Prize in Economics). In 2015, he and Louis Nirenberg were awarded the Abel Prize for their contributions to the field of partial differential equations. As a graduate student in the Mathematics Department at Princeton University, Nash introduced a number of concepts (including Nash equilibrium and the Nash bargaining solution) which are now considered central to game theory and its applications in various sciences. In the 1950s, Nash discovered and proved the Nash embedding theorems by solving a system of nonlinear partial differential equations arising in Riemannian geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
