Finding an appropriate Riemannian metric $G:\mathbb{R}^3\rightarrow\mathbb{R}^{3\times{3}}$ on $\mathbb{R}^3$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:34:23Z http://mathoverflow.net/feeds/question/43617 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43617/finding-an-appropriate-riemannian-metric-g-mathbbr3-rightarrow-mathbbr3 Finding an appropriate Riemannian metric $G:\mathbb{R}^3\rightarrow\mathbb{R}^{3\times{3}}$ on $\mathbb{R}^3$ Majid 2010-10-26T02:10:31Z 2010-10-26T07:49:17Z <p>I am trying to find a Riemannian metric $G:\mathbb{R}^3\rightarrow\mathbb{R}^{3\times{3}}$ on the manifold $\mathbb{R}^3$ such that $G$ is not uniformly positive definite, and there is no isometry $\phi:\mathbb{R}^3\rightarrow\mathbb{R}^3$ satisfying $G=\phi^*I_3$, where $I_3$ is the identity matrix. Moreover, I want $(\mathbb{R}^n,G)$ be geodesically complete. It seems easy to find $G$, but I could not succeed to find any.</p> http://mathoverflow.net/questions/43617/finding-an-appropriate-riemannian-metric-g-mathbbr3-rightarrow-mathbbr3/43643#43643 Answer by Christian Blatter for Finding an appropriate Riemannian metric $G:\mathbb{R}^3\rightarrow\mathbb{R}^{3\times{3}}$ on $\mathbb{R}^3$ Christian Blatter 2010-10-26T07:49:17Z 2010-10-26T07:49:17Z <p>Try $ds^2 = |d{\bf x}|^2/(1+|{\bf x}|^2)$. The resulting metric space is complete since the "horizon" is infinitely far away. But maybe I have misunderstood the notion "not uniformly positive definite".</p>