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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 …

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 …

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 …

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 …

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 …

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 …

Elipsoid does not posess unbounded geodesics with no self-intersection. I do not know a conceptual explanation. My explanation is that (due to integrability of the geodesic flow of ellipsoid) we …

Generic metric does not admit a lot of properties some special metrics admit. A good demonstration of this is the examples listed in the question (no multiple eigenvalues) or in the answers of Matheus …

I assume that imranal asks how to find numerically a geodesic connecting two given points if the connection is given. One way to do it is to implement the solution of the ODE system he wrot …