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Can we say something about it's curvature? Is it true that its sectional curvature must be everywhere non-positive (or at least at some point)? … (If not, can we say something about the Ricci or scalar curvature?) Note that it's clearly not true if we only assume $\exp$ is a diffeomorphism at a single point. …

For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ with non-positive scalar curvature such that $\|g-g_{\epsilon}\|_{C^0}<\epsilon$? … I ask whether the metrics of non-positive scalar curvature are dense in the space of metrics. …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\TM}{T\mathcal{M}}$ $\newcommand{\Ric}{\operatorname{Ric}}$ $\newcommand{\Volg}{\operatorname{Vol}_g}$ I would like to find a reference for the following c …

($J(t)$ is a Jacobi field, $k$ is the sectional curvature of the relevant plane) An immediate corollary (2.10 in the book) is: $\text{(2) } \|J(t) \|=t-\frac 16kt^3+\tilde R(t)$ where $(\frac{\tilde … (Perhaps the values of the sectional curvature in nearby points or its derivatives?) …

We should probably restrict here to manifolds without boundary, since otherwise $M=[0,1],N=\mathbb{R},f(x)=x$ is an example. ) Context: The point is to see whether curvature differences obstruct existence …