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

By Oldrich Kowalski, Eric Boeckx, Lieven Vanhecke

This e-book 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 impressive areas. additionally parallel theories are constructed for manifolds of "relative conullity two". This makes a bridge to a survey on curvature homogeneous areas brought by way of I.M. Singer. As an program of the most subject, fascinating hypersurfaces with variety quantity in Euclidean house are found, specifically these that are in the community inflexible or "almost rigid". The unifying procedure is fixing explicitly specific platforms of nonlinear PDE

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Further put A= fX(f>Go„dx. 3. TWO POSSIBLE SIMPLIFICATIONS 39 Then A is a column matrix function depending only on w and x. Now, introduce new variables ( v 1 , . . , vn) instead of ( y 1 , . . , yn) by the formula 3L = y + X. 45) dv = (d)y + {dy) + d\ d<£ = (G dx + p dw), dX = (G0 dx + pQ dw) where p, p0 are new matrix functions depending only on w and x. 45) we get dv = ((Gy + Go) dx + (py + pQ) dw) + dy. On the other hand, we have the following expression for u : UJ = dy + (Gy + Go) dx + (Hy + H0) dw.

J / n ) in terms of the functions y>} and 7? (z,j = 1,2; a = l , . . , n ) . *• and 7f (i,jf = 1,2; a = 1 , . . , n ) to ensure that the nullity vector space is spanned by ( Y i , . . , Yn) and hence, t h a t the space is foliated by Euclidean leaves of codimension two. These are reduced to a system of non-linear partial differential equations of first order for a certain set of functions of the two variables w\ and W2 only. Moreover, this system can be rewritten as a system of ordi­ nary differential equations with one variable and one parameter.

Yn) dx, a = 1 , . . , n. Here Af ^ 0 and the equations ul = J1 = 0 (or, equivalently, w = constant and x = constant,) determine the Euclidean leaves of dimension n. ( y 1 , . . , yn) gives a Cartesian coordinate system on each leaf Proof Choose a point p of M and a neighbourhood U1 of p. Let S : D2 —■> Uf be a surface through p which is transversal to the Eu­ clidean leaves at all cross-points. Then there exists a normal neigh­ bourhood U of p, U C U', with the property that each point q of U is projected to exactly one point, 7r(g), of S via some Euclidean leaf.

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