(Re)constructing Dimensions

Compactifying a higher-dimensional theory defined in R^{1,3+n} on an n-dimensional manifold {\cal M} results in a spectrum of four-dimensional (bosonic) fields with masses m^2_i = \lambda_i, where - \lambda_i are the eigenvalues of the Laplacian on the compact manifold. The question we address in this paper is the inverse: given the masses of the Kaluza-Klein fields in four dimensions, what can we say about the size and shape (i.e. the topology and the metric) of the compact manifold? We present some examples of isospectral manifolds (i.e., different manifolds which give rise to the same Kaluza-Klein mass spectrum). Some of these examples are Ricci-flat, complex and K\"{a}hler and so they are isospectral backgrounds for string theory. Utilizing results from finite spectral geometry, we also discuss the accuracy of reconstructing the properties of the compact manifold (e.g., its dimension, volume, and curvature etc) from measuring the masses of only a finite number of Kaluza-Klein modes.Comment: 23 pages, 3 figures, 2 references adde

Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds

summary:A Riemannian manifold is said to be semisymmetric if $R(X,Y)\cdot R=0$. A submanifold of Euclidean space which satisfies $\bar{R}(X,Y)\cdot h=0$ is called semiparallel. It is known that semiparallel submanifolds are intrinsically semisymmetric. But can every semisymmetric manifold be immersed isometrically as a semiparallel submanifold? This problem has been solved up to now only for the dimension 2, when the answer is affirmative for the positive Gaussian curvature. Among semisymmetric manifolds a special role is played by the foliated ones, which in the dimension 3 are divided by Kowalski into four classes: elliptic, hyperbolic, parabolic and planar. It is shown now that only the planar ones can be immersed isometrically into Euclidean spaces as 3-dimensional semiparallel submanifolds. This result is obtained by a complete classification of such submanifolds