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Looking on the Minkowski Finlser metric (when the Finsler function does not depend on the point of the manifold) we see that the 4th power of the distance function can not be very smooth. Indeed, f …

Edited: I misunderstood the question. My answer (which is now starts after ``old version of the answer'' below ) answers the following question: suppose there exists a vector field along a curve $c …

Of course not. Scalar curvature is a function only and carries much less information than the metric. For example, take a metric on the sphere such that its scalar curvature has two critical points …

If I am not mistaking, your number is the same for all manifolds, does not depend on the metric and on the manifold and coincides with the packing number of the standard ball in the euclidean $R …

If a manifold equipped with a pseudo-Riemmanian (= nondegenerate but not necessary positively definite) metric contains a region with constant nonzero curvature tensor, then its holonomy group is the …

I hope that my ``answer'' will not be understood solely as a propaganda of my survey http://arxiv.org/abs/1101.2069 where I discussed (1) how, given geodesics, to reconstruct a connection (in both …

We say that two metrics are affinely equivalent if their Levi-Civita connections coincide. Is it possible that an Einstein (=Ricci tensor is proporional to the metric) is affinely equivalent to a metr …

I assume you are asking why two definitions of the projective structures, one given in terms of atlas, and another given as the existence of projectively flat connection, coincide. If two connect …

Let $v$ be a vector field. Does there exists a volume form $\Omega$ such that its Lie derivative is proportional to itself with a constant coefficient: $$\mathcal{L}_v \Omega= C \cdot \Omega? \ \ \ …

On a compact manifold, you have a global frame such that $[X_i, X_j]=0$ if and only if your manifold is the torus. Starting from dimension 3, there are parallelisable manifolds different from the t …

On the torus $T^2$ with the coordinates $x,y$ and the flat metric $g= dx^2 + dy^2$ take any function $f(x)$. Its hessian is given, after raising the index, by the (1,1)-tensor $f''(x) dx\otimes \fra …

In the case the integrals are polynomial in momenta, a generic metric does not poses those (except trivial integrals such as the energy and polynomial functions of the energy). This is a local statem …

By the so-called conformal Lichnerowicz conjecture (proved by Alekseevsky, Ferrand, Schoen) a manifold has either big conformal group or there exists a metric in the conformal class such that the conf …

May be I misunderstood your question; I reformulate it as follows: whether there exists a regular embedding of a complete surface of constant positive curvature in $R^3$ and whether this surface can …

Most (in the natural sense) connections have the same holonomies, namely the maximal one. Most affine connections have the same holonomy, namely the whole $GL_+(n)$. Most Levi-Civita connections hav …