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To add to Sam's answer, compatible almost complex structures are often needed in applications of Gromov-Witten theory to Hofer geometry. Even more fundamentally they are necessary in the construction …

For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by: $$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ^ …

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space o …

Let $M$ be a closed manifold with non-torsion $\pi_2$, and $A$ a non-trivial free homotopy class of a map $f: S^2 \to M$. Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean curva …

It is a well known theorem of Anderson that any vector bundle over a negative sectional curvature Riemannian manifold admits a metric of negative sectional curvature. Q: Are there examples of complete …

The answer is No. Indeed it is no for spheres of all dimension assuming one can find an immersed class $A$ $S^n$ with mean curvature less than a given $C(A)$. The counterexamples can be generalized f …

It is known since Bochner that for M compact with negative Ricci curvature, the group of isometries is discreet and hence finite. Are there any generalizations to compact non-positive sectional curvat …

Is the following true? I cannot see a counterexample and it seems very intuitively clear, at least in the embedded case. Claim: Consider the set $S$ of closed immersed Riemann surfaces $\Sigma \subse …

Suppose X,g is a complete negative sectional curvature Riemannian manifold. And $C \subset X$ is compact submanifold. What are the minimal conditions on $C$ so that we can deform g on the restriction …