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I did not understand the first question Question 1 Are there manifolds with the property that each connection on is never flat? Because one of course can construct, on any manifold, a connectio …

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 …

The answer is ``no'', the pointwise condition is not enough. The example exists in dimension 1 already and can be generalized and made arbitrary weird for all dimensions. Consider a smooth functi …

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? \ \ \ …

Of course, yes. Lie derivative is defined for any geometric object (= when it is defined what happends when we change a coordinate system): take the flow $\phi_t$ of the vector field, consider the …

Let us first deal with linear algebra. Assume a matrix $J$ satisfies $J^k= -Id$. Then, there exists a poylnomial $P$ whose coefficients depend on the eigenvalues of your $J$ such that $P(J)$ is a …

The answer of j.c. given prior to mine is of course correct but let me give a trivial reason why (in big dimensions) every Riemannian metric after an arbitrary small perturbation is not isometr …

No, because otherwise we will have this property also for degenerate ellipses, which are intervals, which would imply that the euclidean distance between two (sufficiently close) points is $\lambda …

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 …

The answer depends on the diffeomorphism. Let me give two examples, both on the standard torus $\mathbb{R}^2/_{\mathbb{Z}^2}$ with coordinates $x,y$. (Example 1:) $$\phi(x,y)= (x+ 1/2,y).$$ Fo …

You function $d_p$ is a Lipschitz function (w.r.t. to the Riemannian distance) with the Lipschitz constant $1$ and it is possibly, for every $\varepsilon>0$, to $\varepsilon$-approximate it by a smoot …

It is known (say, Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans.Amer.Math.Soc. 117(1965) 251– 275; I am not sure that Tashiro is the first who proved it and there were many …

I give a geometric explanation of the calculations of Willie, which simultaneously elaborates the suggestion of Deane. The flow of a vector field commuting with Laplacian preserves the Laplacian an …

Let me give a standard example of a closed incomplete manifold with flat affine structure, whose 2-dimensional version is essentially the example from the comment of Misha. Consider $R^n\setminus 0$ …

The Riemannian distance has the following property: it is the so called length space. I recall (one of) the definitions of the length space. Having a distance function $d$, we can define the length o …