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It turns out that the answer is 'No, the fourth power of the geodesic distance from a point $p$ in a Finsler space $M$ whose norm, raised to the fourth power, is a smooth convex quartic form is not ge …

Compact surfaces with non-negative Euler characteristic cannot carry such metrics because of Gauss-Bonnet. However, this is the only obstruction. First, such metrics exist on any compact orientable …

Well, I don't know what you mean by 'natural coordinates', so I might be entirely off, but here's a possible interpretation of what you are asking about: First, assume given an $n$-manifold $M$, a Ri …

This is an elementary exercise in the differential geometry of curves and really belongs in MathStackExchange. Assume that the curve is parametrized by arclength, say $X(s)$, and that your given ve …

I believe that this is discussed in Zhongmin Shen's book "Lectures on Finsler Geometry". The result, if I remember correctly, is that the square of the geodesic distance function is generally not smo …

There is such a list, but I don't know where it might already be written down. It's not hard to compile it, though, using known facts about the classification of symmetric spaces. First, one can eas …

I will explain how you can get the formulae for the closed $2$-forms that define the hyperKähler structure without introducing coordinates. Once you know the $2$-forms, you can find the $I$, $J$, and …

If you assume that $M$ itself is connected, then it is true. First, if $M$ is orientable and connected, then $O(M)$ has (at least) two connected components, one for each orientation of frame, either …

You might need to be more precise in your question. As BS noted there exist Zoll $2$-spheres that are not surfaces of revolution, but you may have wanted to know whether there are any explicitly know …

Yes, there are many such examples. A totally real submanifold in this case is simply a surface $S\subset\mathbb{R}^4$ such that, for each $p\in S$, the space $J(T_pS)$ is orthogonal to $T_pS$. In …

No. Consider a curve on the unit $2$-sphere, parametrized by arc length $ds$ with geodesic curvature $\rho(s)$. One easily computes that, as a curve in $3$-space, one has $$ \kappa(s)^2 = 1 + \rho …

I'll give a partial answer to the OP's second question, which I take to be asking for the obstructions for a metric in dimensions greater than $2$ to be conformal to a product metric. This is, firs …

There may be obstructions, even in the projective case and even when the vector field does not vanish to second order. Suppose given a projective structure on an $n$-manifold $M$ and a projective vec …

Working this out in full generality is likely to be rather messy and unenlightening, but there are a few remarks about the nature of the problem (and special cases) that can be made, so I'll put them …

I assume that, by 'corresponding frame' for a given chart, you mean the local frame field defined by the coordinate vector fields of the chart. If that is not what you mean, you should specify. Also …