Download Critical Point Theory and Submanifold Geometry by Richard S. Palais PDF

By Richard S. Palais

Show description

Read Online or Download Critical Point Theory and Submanifold Geometry PDF

Best differential geometry books

Surveys in Differential Geometry: Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer (The founders of the Index Theory) (International Press) (Vol 7)

The Surveys in Differential Geometry are supplementations to the magazine of Differential Geometry, that are released through overseas Press. They comprise major invited papers combining unique learn and overviews of the most up-tp-date study in particular parts of curiosity to the turning out to be magazine of Differential Geometry neighborhood.

Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics

Vital transforms, akin to the Laplace and Fourier transforms, were significant instruments in arithmetic for no less than centuries. within the final 3 a long time the improvement of a couple of novel rules in algebraic geometry, classification concept, gauge idea, and string thought has been heavily with regards to generalizations of necessary transforms of a extra geometric personality.

Riemannsche Geometrie im Großen

Aus dem Vorwort: "Globale Probleme der Differentialgeometrie erfreuen sich eines immer noch wachsenden Interesses. Gerade in der Riemannschen Geometrie hat die Frage nach Beziehungen zwischen Riemannscher und topologischer Struktur in neuerer Zeit zu vielen sch? nen und ? berraschenden Einsichten gef?

Geometric analysis and function spaces

This ebook brings into concentration the synergistic interplay among research and geometry by means of analyzing quite a few themes in functionality idea, genuine research, harmonic research, a number of complicated variables, and staff activities. Krantz's strategy is stimulated through examples, either classical and smooth, which spotlight the symbiotic courting among research and geometry.

Extra info for Critical Point Theory and Submanifold Geometry

Example text

B y assumption w e have wi. = f(x)wi. 4) 2. 13), we obtain dwi~ = df A wi + f E wij A a~j J = E fjcoj A COi "J- f E cOij A ~Oj J J J J So ~ j f j w i A ~ i = O, which implies that f j = 0 for a l l j # i. , f = c a constant. If c = O, then wi,~ = 0. , X ( M ) is contained in a hyperplane. If c 7{ O, then coi~ = cwi and d X + = E |• iei -- --03ia ¢ ei--=0. So X + e~/c is equal to a constant vector xo E R n+l, which implies that IIX - x01t 2 = ( 1 / c ) 2. 10. Definition. An immersed submanifold M n of the simply connected space form Nn+k(c) is called totally umbilic if I I = ~1, where ~ is a parallel normal field on M .

Wi,n+l = cotOwi and I I = c o t 0 I . g. in S n+l, if cos 0 = 0 (or equivalently V is a linear hyperplane). 8. E x a m p l e . and Submanifold Theory Let v0 be a non-zero vector o f the Lorentz space M= {x e R ' ~ + l ' l I ( x , x ) = - 1 , R n+l'l, (X, V o ) = a } . 2) for s o m e b. Note that (vo, vo) = --a 2 + b2. 2), we have ~-~i(awi + k~i,,~+l)ei = 0. So a w i + bcvi,n+l = 0. 3) (i) If (vo,v0) = 1, then - a 2 + b2 = 1 and we may assume that a = s i n h t o and b = cosh to. , M is totally umbilic with sectional curvature - 1 + t a n h 2 to = - s e t h 2 to.

PROOF. It follows from the definition of O3AB that w = (WAB) is an o(n + k ) - v a l u e d 1-form on M. 4) imply that w satisfies Maurer-Cartan equation: d~z7 = ~v A ~ , which is the integrability condition for the first order system d~ = w~2. 40 Part I Submanifold Theory So there exist a small neighborhood U of :Co in M and maps eA : U ~ R "+k such that deA = E a C A B @ eB, B where eA(xO) = VA and {eA(Z)} is orthonormal for all z E U. 5) is zero and hio~j = hjcd implies the second term is zero.

Download PDF sample

Rated 4.82 of 5 – based on 14 votes