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Register a new userGeneralized (kappa,mu)-space forms
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Ver\\'onica Mart\\'in-Molina
Publication date
2012
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English
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Generalized (kappa ,mu)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case.
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S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its perturbed equation and prove that the space of volume forms is an infinite dimensional non-positively curved metric space in the sense of Alexandrov.
We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $mgeq 3$ of a product $Q_{k_1}^{n_1}times Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2 eq 0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of $Sf^ntimes R$ and $Hy^ntimes R$. The case of $Sf^ntimes R$ was carried out in cite{mt}. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.
We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $fcolon M^n_ctoQ^{n+p}_{tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature $tilde c$, and free of weak-umbilic points if $c>tilde{c}$. We show that the substantial codimension of $f$ is $p=n-1$ if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank $n-1$. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit B{a}cklund and Ribaucour transformations.
In this paper we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson-Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.
We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.
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