# Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

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### In a normal coordiantes, the determinant of the metric tensor can be viewed as a function of the radial distance function? ( And furthur question )

I'm reading the John Lee's Introduction to Riemannian manifold, p.333, p.335 and some question arises First of all, let's see Lemma 11.13 in his book p.333 Lemma 11.13. Suppose $(M,g)$ is a ...
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### Why is the set of singular points of starlike boundary $\Gamma$ closed?

I'm reading Geometric Inequalities of Yu. D. Burago and V. A. Zalgaller. I don't understand why $E_1$ is closed in the proof of the following lemma. Several definition. Suppose $\Omega$ is a ...
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### Levi-Civita connection as an Ehresmann connection

I've been trying to learn how to define the Levi-Civita connection as an Ehresmann connection, and I am getting quite confused. hoj201's answer on https://mathoverflow.net/questions/34088/a-geometric-...
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### A consequence of Huisken's rescaled monotonicity formula : Stone's Lemma

The context is that of the mean curvature flow, more precisely, concerning Type I singularities and the rescaling procedure. The text I am following is by Mantegazza: "Lecture notes on Mean ...
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### Understanding the Gunther's Volume comparison theorem ( John Lee's Introductino to Riemannian manifold ) [closed]

I am reading the John Lee's Introduction to Riemannian manifold, p.334, Theorem 11.14 and stuck at quite a few point. I am self studying Riemannian geometry and it seems that this proof is one of the ...
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### Subharmonic in a Neighbourhood of a critical point

$M$ is a compact Riemannian manifold and $f$ is a smooth function with the property that for each critical point there is a neighborhood in which $f$ is subharmonic. Can we say that $f$ is subharmonic ...
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### Inversive product of tow ultraparallel geodesics in the hyperbolic plane is $\cosh{\rho}$

This is Lemma 7.17.3 in Beardon: Lemma 7.17.3 Let $L$ and $L'$ be geodesics in the hyperbolic plane. Then the inversive product $(L,L')$ is $\cosh{\rho(L,L')},1,\cos{\phi}$ according as $L,L'$ are ...
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### Computing tangent vector for sphere

I am not sure if I'm taking the derivative of a transition function correctly. This is coming from exercise 1.28 of Riemannian Geometry by Gallot-Hulin-Lafontaine. Given the sphere $S^2$ and two ...
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### Reference for currents in geometric measure theory - continuous version?

Let $M$ be a Riemannian manifold. I am interested in the dual of the space of continuous (not necessarily smooth) vector field on $M$. The dual of the space of smooth vector field would be the space ...
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### Does a coercive Riemannian metric imply geodesic completeness?

Let $M \subset \mathbb{R}^n$ be a bounded open (in the euclidean sense) set. Define a smooth function $Q:M \to S_{++}^n$, where $S_{++}^n$ denotes symmetric positive-definite $n \times n$ matrices. ...
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### Isometries of the hyperbolic plane

$\color{brown}{\text{I was reading about}}$ the Mobius transformations $\mathcal{M}$ of the hyperbolic plane $\mathbb{H} := \{a + bi: b > 0\} \subset \mathbb{C}:= \mathbb{R}^2$ and its isometries ...
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### How to construct a $C^1$ variation of the unique, minimal geodesic to its first conjugate point

This is a question that I encountered when reading Proposition 1.8 of the paper "Optimal transport and curvature" by Alessio Figalli and Cedric Villani, whose proof is only outlined and ...
1 vote
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### What is the mean of "the isometries that identify the sides of polygon"?

Picture below is from the 167th page of do Carmo's Riemannian Geometry. I don't know the mean of "the isometries that identify the sides of $P$". My English is poor. I know the process of ...
1 vote
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### What is the connection between the gradient acting on the support function in Euclidean space and on a sphere?

The support function defined on the unit sphere can be locally represented as a support function defined in the entire space. This can be achieved by using a mapping \begin{align} (x_1,\cdots,x_n)&...
1 vote
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### Deriving the round metric on $\mathbb{S}^2$

Consider $\mathbb{S}^2$ as a submanifold of $\mathbb{R}^{3}$. Given that $\mathbb{R}^{3}$ has the standard Riemannian metric $g$, I want to derive a local expression or the metric tensor for the round ...
1 vote
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### There are smooth vector fields along a curve forming a basis of the tangent space at each point

An application of parallel transport is to prove the existence of smooth frame along a curve on the manifold. But it seems that the existence has nothing to do with the metric, so I would like to find ...
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### Big picture: Geodesics on a spherical surface embedded in $\Bbb R^3$

Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$ Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in ...
1 vote
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### Parallel field and splitting of Riemannian metric

Let $(M,g)$ a $n$ dimensional Riemannian manifold, $U \subset M$ an open subset and $E \in \mathfrak{X}(U)$ a unit parallel vector field (i.e.$\nabla{E}\equiv0$ and $g(E,E)\equiv1$). Now consider the ...
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### Scalar curvature and second fundamental form

If one has a Riemannian manifold $M^3$ which is embedded in a $4$-manifold $N^4$ such that $M^3$ has metric $g$ and second fundamental form $k$, does having scalar curvature $R_g$ always larger than ...
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### Analogue of Maximum principle in analytic varieties

I have been reading Tasty bits of several complex variables https://www.jirka.org/scv/scv.pdf . In Section 6.7, we have the following exercise: Exercise 6.7.10 Suppose $X \subset U$ is an irreducible ...
1 vote
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### Geodesics connecting two points in the hyperboloid model of the hyperbolic plane

Let $\mathbb{H}^2$ be the upper sheet of the hyperboloid defined by $x^2+y^2-z^2=-1$ in three-dimensional Minkowski space $(\mathbb{R}^3, g_M)$, where $g_M = \text{diag}(1,1,-1)$. In other words, ...
1 vote
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### Roadmap for learning symplectic geometry, starting from differential geometry

I am a Physics Major. I have done an undergraduate level course of differential geometry. I want to get into symplectic and Riemannian geometry, but I would like to start over from differential one ...
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### (locally) symmetric spaces where every conformal transformation is an isometry

By an argument using Liouville's theorem for conformal maps Conformal automorphism of $H^n$ it can be shown that every conformal automorphism of a hyperbolic manifold is an isometry. Are there any ...
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### Is the group of conformal automorphisms of a finite volume hyperbolic manifold finite?

Let $M$ be a Riemannian manifold. Let $Iso(M)$ be the group of isometries of $M$. Let $Conf(M)$ be the conformal group of $M$ (the group of all diffeomorphisms of $M$ that preserve the ...