Download e-book for kindle: Riemannian Manifolds of Conullity Two by Oldrich Kowalski, Eric Boeckx, Lieven Vanhecke

By Oldrich Kowalski, Eric Boeckx, Lieven Vanhecke

ISBN-10: 981022768X

ISBN-13: 9789810227685

This booklet bargains with Riemannian manifolds for which the nullity area of the curvature tensor has codimension . those manifolds are "semi-symmetric areas foliated through Euclidean leaves of codimension " within the feel of Z.I. Szabo. The authors be aware of the wealthy geometrical constitution and particular descriptions of those extraordinary areas. additionally parallel theories are built for manifolds of "relative conullity two". This makes a bridge to a survey on curvature homogeneous areas brought by means of I.M. Singer. As an software of the most subject, attention-grabbing hypersurfaces with kind quantity in Euclidean house are came upon, specifically these that are in the neighborhood inflexible or "almost rigid". The unifying procedure is fixing explicitly specific platforms of nonlinear PDE

Show description

Read Online or Download Riemannian Manifolds of Conullity Two PDF

Similar differential geometry books

New PDF release: Differential Geometry with Applications to Mechanics and

Compiling facts on submanifolds, tangent bundles and areas, critical invariants, tensor fields, and enterior differential kinds, this article illustrates the elemental techniques, definitions and houses of mechanical and analytical calculus. additionally bargains a few topology and differential calculus. DLC: Geometry--Differential

J. Śniatycki's Differential Geometry of Singular Spaces and Reduction of PDF

During this publication the writer illustrates the ability of the idea of subcartesian differential areas for investigating areas with singularities. half I provides a close and entire presentation of the idea of differential areas, together with integration of distributions on subcartesian areas and the constitution of stratified areas.

New PDF release: Integral Geometry And Tomography: AMS Special Session on

This quantity includes a set of papers that brings jointly primary examine in Radon transforms, indispensable geometry, and tomography. It grew out of the particular consultation at a Sectional assembly of the yankee Mathematical Society in 2004. The e-book comprises very contemporary paintings of a few of the pinnacle researchers within the box.

Additional resources for Riemannian Manifolds of Conullity Two

Sample text

We construct a positive definite Riemannian metric on the set of vectors X for which (X,X) < 0 as follows. For a negative number —a2 the vectors X sat­ isfying (X,X) = —a2 form a hypersurface Hn(-a2). 3. RIEMANNIAN MANIFOLDS OF CONULLITY TWO 19 definite inner product which defines a hyperbolic metric on h Hn(-a2). Let us consider this positive inner product (•, •) on the tangent spaces of the various hypersurfaces Hn(-a2). For a vector X pointing from a point p e Hn(-a2) to the origin o we define (X,X) = -62(X,X), (X,Y) = 0, for aU Y € TpHn(-a2), for some fixed positive constant b2.

Clearly, this then holds for all the factors in the local decomposition of (M, #). But the scalar curvature of any type of cone in Szabo's structure theorem is non-constant, so the local decom­ position of (M,g) does not contain factors of cone type. Moreover, a two-dimensional manifold with constant scalar curvature is locally iso­ metric to a two-dimensional symmetric space. Hence, (M,g) is locally a product of symmetric spaces and of Riemannian manifolds foliated by totally geodesic Euclidean leaves of codimension two with constant scalar curvature.

N +2) the orthonormal frame which is dual to the coframe (a; 1 ,.. ,a; n + 2 )). This function is never zero because the nullity index is, by definition, equal to n. The components Ct\i ^iL+2> ^a+2 a n d ^/3+2 °f ^he c u r v a ture form (with respect to the coframe (a; 1 ,.. ,u; n+2 )) must satisfy Ct\ = kul o 1 - o Au2, 2 - oa+2 - n fl$ + flj = 0, z , j = l , . . , n + 2. 9) 1 2 = 0, = a; A a; A a;£+2 0, = 0. 11) _ ( A / A ) = 0, Afk = a= &(w,x)^0. 1. A CANONICAL FORM FOR THE METRICS 27 Next we write u\ = a^ui1 + a\2w2 + Y, «2 a+2 w0r+2 ) a P "1+2 = C a+21^ 1 + 4+22^ 2 + E C a + 2 / ?

Download PDF sample

Riemannian Manifolds of Conullity Two by Oldrich Kowalski, Eric Boeckx, Lieven Vanhecke


by Kevin
4.0

Rated 4.76 of 5 – based on 44 votes