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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.
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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 give local and global parametric classifications of a class of Einstein submanifolds of Euclidean space. The highlight is for submanifolds of codimension two since in this case our assumptions are only of intrinsic nature.
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.
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.
Let $nge 2$ and $kge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than $delta<0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue $lambda_1(M)$ of $M$ satisfies $$lambda_1(M)le nleft(delta +frac{1}{operatorname{Vol} M}int_M |H|^2 operatorname{dvol}right).$$ The equality implies that $M$ is minimally immersed in a metric sphere after lifted to the universal cover of $N$. This completely settles an open problem raised by E. Heintze in 1988.
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