By Oldrich Kowalski, Eric Boeckx, Lieven Vanhecke
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
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Additional resources for Riemannian Manifolds of Conullity Two
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 / ?
Riemannian Manifolds of Conullity Two by Oldrich Kowalski, Eric Boeckx, Lieven Vanhecke