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Is there an isometric embedding of the modular surface $X(1)=PSL(2,\mathbb{Z})\backslash \,\mathbb{H}$ into the Euclidean 3-space? For all I know this may be an open problem but I am also curious if a …

The answer is already given in the comments (by Ryan Budney and Mizar). But I think it makes sense to clear this confusing point. The classical Gauss-Bonnet formula is [e.g. https://en.wikipedia.or …

Consider a Riemannian manifold $\Sigma$ of dimension two homeomorphic to a torus. When is there a non-closed geodesic on $\Sigma$ which does not intersect itself — are there reasonable necessary or s …

Let $(M,g)$ be a Riemannian manifold; for the sake of simplicity we assume that its group of isometries is trivial. If we consider the same manifold equipped with another metric $g'$, what is the asym …

It appears to be a standard fact in topology that $\mathbb{C}\mathbb{P}^2\#-\mathbb{C}\mathbb{P}^2$ has a structure of a $\mathbb{S}^2$ bundle over $\mathbb{S}^2$. Is there a nice geometric descripti …

Take a torus $\mathbb{T}^d$ of dimension $d$ and introduce on it such a metric that some part of it would be isometric to a (sufficiently small) neighbourhood $U$ of $p\in M$. By Hodge theory, harmon …

I believe that no ``natural'' section of either $E^*\otimes E^*\otimes E$ or $T^*\otimes T^*\otimes E$ or even $T^*\otimes E^*\otimes E$ can be associated with a general connection on a vector bun …

Let $X$ be a compact hyperbolic surface, and $\gamma$ a closed geodesic on it. Consider the integral $$\int_\gamma f(x)\, dl(x)$$ where $f$ is a (normalized) Laplace eigenfunction on $X$. Intuiti …

Let $X$ be a complex manifold, and $Y\subset X$ a compact submanifold. Is it true that the tangent bundle $TY$ may be extended (as a holomorphic vector bundle) to some open neighbourhood of $Y$ in …

The differential of the exponential map on a symmetric space can be expanded (abusing some notation) as $d{\rm Exp}_X=\sum_{n=0}^{\infty}\frac{({\rm ad}X)^{2n}}{(2n+1)!}.$ This is an old (1958) res …