Colloquium abstracts

Mahuya Dutta
I.S.I., Kolkata
September 8, 2011

Lipschitz Isometric Maps for Pairs of Riemannian Metrics:  Nash proved in 1954 that any Riemannian manifold $(M,g)$ of dimension $n$ admits an isometric $C^1$-immersion into the Euclidean space $\mathbb R^q$, provided it admits a $C^\infty$-immersion in $\mathbb R^q$ for some $q \geq n+2$. In 1955 Kuiper improved this result by relaxing the dimension condition to $q\geq n+1$. The technique of Kuiper was later adapted into the theory of Convex Integration by Gromov. We shall discuss some genralisations of Nash-Kuiper theorem for pairs of Riemannian metrics on the manifolds using the convex integration technique.\\ \textsc{Main Result.} Let $h_1,h_2$ be two Euclidean metrics on $\mathbb R^q$, such that for any $c\in(a,b)$, $c^2h_1-h_2$ is an indefinite metric with signature $(r_+,r_-)$, where $r_\pm\geq 2n$. If $M$ is manifold of dimension $n$ with a pair of Riemannian metrics $(g_1,g_2)$ related by the inequalities $a^2g_1< g_2