# Tagged Questions

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### Holomorphic coordinates on a Kähler manifold

Let $(X,J,\omega)$ be a Kähler manifold. Let $\dim_{\mathbb{R}}(X)=2n$ and we also know that $X$ splits as $X=M\times \mathbb{R}$, where $\dim_{\mathbb{M}}=2n-1$. My question is now: does there exists ...
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### Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements $\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...
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### Area on the unit sphere swept out by big circles corresponding to a curve

For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the ...
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### Existence of Simple Closed Straightest Geodesics

There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov]. Is the same true ...
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### Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $\lambda > 0$) and the ...
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### Optimal paintbrush geodesics

Let $S$ be a smooth, closed surface in $\mathbb{R}^3$, and $\gamma$ a geodesic segment on $S$, i.e., a finite-length piece of a geodesic. Define $\gamma(w)$ as all the points of $S$ within a distance ...
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### Constructing a special infinite-dimensional vector bundle

Let $M$ and $N$ be finite dimensional smooth manifolds and $p: E \rightarrow N$ a finite rank vector bundle. $M$ can be assumed to be compact if necessary but I would prefer to work without this ...
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### Lorentzian isoparametric hypersurfaces of ads

A hypersurface of a pseudo-Riemannian manifold is said to be isoparametric if its shape operator has the same characteristic polynomial at all points. Xiao has classified Lorentzian isoparametric ...
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### About the parallel transport and choice of connection

Thought Experiment Consider a 2-sphere, $S^2$, and let $p$ be a point at the equator. Case 1 Let us parallel transport a vector, $V$ from $p$ using the recipe: Move one unit of length East. Move ...
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### Are all null curves of a Lorentzian metric extrema?

Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of $$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$ The so-called "null curves" are ...
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### Tangent lines to 2 circles, tangent planes to 3 spheres, and so on.

Although it is known the solution to the first two questions, somebody may have different nice answers, so I include them: Given two circles in the plane, there is (at least) a line which is tangent ...
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### degenerating surface II

In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is ...
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Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help. We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...
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### Probing a manifold with geodesics

Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$ embedded in $\mathbb{R}^3$. You do not know anything about $M$. You shoot off a geodesic $\gamma$ in some direction $u$, and learn ...
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### Is there an ellipsoid with given outer normals?

Pick two points $(x,0)$ and $(0,y)$ (say $x>0$ and $y>0$). Pick a unit vector $u = (u_1,u_2)$, $v = (v_1, v_2)$, and attach one to each of the points. Provided $u$ and $v$ are "nice" ($v$ needs ...
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### Constructing a hypersurface with given outer normals

Pick a point on each of the positive half-axes in $\mathbb{R}^n$. Put a (unit-norm?) vector at each of the n points. (a) Is there a hypersurface in the orthant $\mathbb{R}^n_+$ going through these n ...
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### Why is the half-torus rigid?

The half-torus surface that results from slicing a torus like a bagel, depicted below (left), is isometrically rigid.       I know this from a remark of Alexandrov in Mathematics: ...
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### Relationship between the focal locus and the cut locus

I am seeking clarification of the relationship between the focal locus and the cut locus of a curve $C$ in $\mathbb{R}^2$, and of a surface $S$ in $\mathbb{R}^3$. Essentially my question is, Under ...
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### Developable 3-manifolds in $\mathbb{R}^4$

Is there a classification of the equivalent of a "developable surface" in $\mathbb{R}^4$? Analogous to: planes, cylinders, cones, and tangent developables in $\mathbb{R}^3$? Edit: Here I am imagining ...
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### Shortest paths on linked tori

I will make this question specific at first, and general later. Suppose we have two linked tori, $T_1$ and $T_2$, each of radii $(2,1)$, meaning that each torus is the result of sweeping a circle of ...
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### $G$-structures of finite type.

A $G$-structure $\pi : B_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie ...
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### Eigenvalues of Laplacian-Beltrami operator

I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value. I am interested in the case when ...
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### Geometric interpretation of Cartan's structure equations

Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^*M$ we can define the connection 1-forms $\omega^j_i$. We define the curvature 2-forms by ...
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### analogues of Cayley plane as homogenous spaces

The Cayley projective plane $\mathbb{OP}^2$ can be defined as a homogenous space $\mathrm{F_4/Spin(9)}$, where $\mathrm{F_4}$ is the compact exceptional simple Lie group. The other possible approach ...