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If we have a Killing tensor field $K$ of type $(0,d)$, the function $$I:SM\to \mathbb{R}, \ I(v)= K(v,\dots, v) \ \ \ \ \ \ (\ast )$$ is constant along geodesic flow. This is a well-known knowled …

It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point, and any two othogonal vectors in this plane, and any neg …

Any conformal map in dimensions $\ge 3$ is necessary a superposition of inversions and isometries (see e.g. the link suggested by Daniele Tampieri in his comment), so it takes the boundary of $D_1$ to …

Any symplectic 2n-dimensional manifold admits a systems of Poisson-commuting functions $f_1,..,f_n$ whose differentials are linearly independent on an open subset of full measure. The result is prove …

If a manifold is complete, the existence of the function $\phi$ such that $\nabla_i \nabla_j\phi = g_{ij}$ implies that the metric is flat and that in a `flat' coordinate system such that the metric …

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? \ \ \ …

I confirm the Anton's answer (No, and the phenomenon is essentially local), but I suggest another explanation which works for C^1 2-dimensional metrics. We will look for a counterexample in the clas …

Under stronger regularity assumptions, an analog of the curvature exists in the weak sense, i.e., in the sense of generalized functions. The stronger regularity assumption is that the metric (in your …

The picture above gives a counterexample to your hope. It is in dimension 2 but there is no problem to make it in any higher dimension. The commuting vector fields are red and blue, the compact is n …

Exponential map in your definition is closely related to the smooth family of smooth curves smoothly depending on the position such that in every point in every direction there exists precisely one …

The generator of the diffusion corresponding to a Riemannian metric (i.e., the diffusion process which the limit of the random walks such that their increments go along geodesics and the distributi …

The rectification theorem says that near a regular point every vector field $v$ is standard, that is, there exists a local coordinate system such that $v=\frac{\partial }{\partial x_1}$. In the textb …

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 …

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 …

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 …