Questions tagged [riemannian-geometry]
For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.
7,578
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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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Time derivative of squared distance function under evolving metric
Suppose we have an evolving familly of Riemannian manifolds $\{M, g(t)\}_t$ indexed by time $t$. Typically a flow. We fix a point $x$ in $M$ and consider the following function : $$ f_x(t;y) = d_{g(t)}...
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How to show that there must be a point on a compact smooth torus in $ \mathbb{R}^3 $ with positive or negative Gauss curvature?
Assume that $ T $ is a two dimenional compact smooth torus in $ \mathbb{R}^3 $. How to show that there must be some point $ p\in T $ such that $ K(p)>0 $ or $ K(p)<0 $, where $ K(p) $ means the ...
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Length of curve on $(S^2,\tilde g)$
Under the polar coordinate, the unit sphere is
$$
S^2=\{(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)\in \mathbb R^3:\theta\in[0,\pi],\varphi\in[0,2\pi] \}
$$
consider a non-induced metric, ...
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How to calculate the area of part of $(S^2,\tilde g)$?
Under the polar coordinate, the unit sphere is
$$
S^2=\{(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)\in \mathbb R^3:\theta\in[0,\pi],\varphi\in[0,2\pi] \}
$$
the induced metric is
$$
g=\...
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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-...
- 300
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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 ...
- 2,028
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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 ...
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Can a Riemannian metric be defined in terms of the cotangent space?
I have always thought of Riemannian metrics as being an inner product assigned to each tangent space. That is, if $M$ is a manifold, then at any point $p \in M$,
$$g_p: T_pM \times T_pM \rightarrow \...
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How do normal coordinates take effect in the proof of evolution equations?
Though the title includes "(geometric) evolution equations", my question is really more of how to use normal coordinates to help us prove an identity involving components of a tensor. My ...
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Lifting a conformal metric obtained by filling a punctured Riemann surface
I have a question concerning lifts/descents of conformal metrics on Riemann surfaces after filling in a puncture. This is a soft question. I'll illustrate my confusion with an example.
Suppose $f:\...
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Is parallel transport or a connection needed for geodesic computation?
I have been reading a bit more about differential geometry and I'm interested in it from the practical computational perspective. I have seen some places where it is mentioned that you cannot move ...
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Compute the laplacian of the distance function in radial coordinates.
Let $M$ be a $n$-dimensional Riemannian manifold with metric $g$ in radial coordinates
$$g=dr^2 + r^2 g_{ij} d\theta_i d\theta_j$$
then
$$\Delta r = \frac{n-1}{r} + \frac{\partial}{\partial r} \log \...
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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 ...
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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)&...
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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 ...
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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 ...
- 1,335
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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 ...
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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 ...
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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, ...
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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 ...
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Proof that geodesics have constant speed [duplicate]
Let $(M,g)$ be a Riemannian manifold. Let $(\varphi,U)$ be a chart. A curve on that chart $\gamma(t)=\varphi^{-1}(x^1(t),...,x^n(t))$ is a geodesic if it solves the geodesic equation $$\ddot{x}^i+\...
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Some confusion about Riemannian symmetric space
Recently I'm learning some basic theories about symmetric spaces from "Differential geometry, Lie groups, and symmetric space" written by Sigurdur Helgason, and I have some confusion. I Hope ...
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Define $\langle,\rangle_{(x,y)} = dx\otimes dx + h(x)^2dy\otimes dy.$ Compute the Gaussian curvature of $\mathbb{R}^2$ with this metric.
Let $h$ be a $C^\infty$ function on $\mathbb{R}$. At each $(x,y) \in \mathbb{R}^2$ define $$\langle,\rangle_{(x,y)} = dx\otimes dx + h(x)^2dy\otimes dy.$$ Compute the Gaussian curvature of $\mathbb{R}^...
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Well-definedness of $\exp_\tilde p \circ i\circ \exp_p^{-1}:S^n-\{q\}\rightarrow \tilde M$
I am reading the 4.1 Theorem of do Carmo's Riemannian Geometry (as pictures below). The well-definedness of red line is not obvious for me.
In fact, I can accept that $\exp_p^{-1}$ map $S^n-\{q\}$ to $...
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Why is the following commuting rule true when working in normal coordinates?
Let $v$ be a smooth function on a manifold and $R$ denote the Riemann Curvature tensor. If we work in normal coordinates, i.e. Christoffel symbols are zero why is the following formula true
$\nabla_i \...
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What does it mean to induce a Riemannian metric on an evolving hypersurface immersed in a Riemannian manifold?
Oftentimes in some journal articles, I encounter a statement like "Let $F:M^n\times[0,T]\to N^{n+1}$ be a one-parameter family of immersions in a Riemannian manifold $(N,g)$ and let $g_t$ be the ...
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A twist on the Hand to Sensor Calibration problem for Riemann metrics
I have two sub-manifolds $M_1$ and $M_2$ of rank $k_1$ and $k_2$. Now given $k_1 \neq k_2$ there is no unique torsion free compatible connection between points in $M_1$ and $M_2$. However, I would ...
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Is the Ricci tensor controlled by the metric? [closed]
Does an estimate of the form
$$Cg(X,X)\leq \operatorname{Ric}(X,X)\leq \frac{1}{C}g(X,X)$$
hold for all $X$ under appropriate assumptions? Thanks.
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Solving the geodesic equation with Sobolev space techniques
I was wondering if there is a proof that every Riemannian metric on $\mathbb{R}^n$ is complete using Sobolev space techniques. Here is the start of an argument. Classically, one can reduce the ...
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Reference request: $L^p$-spaces of vector bundles
I am looking for a good reference on the measure theoretic construction of the spaces $L^p(M;E)$ for a vector bundle $\pi \colon E \rightarrow M$ on a Riemannian manifold $(M,g)$, by which I mean ...
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Help with Parallel transport of a vector
I have a question about parallel transport of a vector.
In picture A) we parallel transport a vector on a (red)curve in a flat plane. This can be clearly seen as the (blue)vector is constant in ...
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Simple closed curve of positive curvature that is stabilized under some rotation
Consider the following statement:
Suppose that $C$ is a smooth simple closed curve in $\mathbb R^2$ with positive curvature. Let $O$ be the mass center of the region bound by $C$. If $C$ is invariant ...
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Proving the generalized Riccati Equation for the second fundamental form
Let $(M,g)$ be an $n$-dimensional Riemannian manifold and $u \in C^\infty(M)$ so that whenever $|\nabla u| \not = 0$, we may write the metric , by the generalized Gauss' Lemma, as
$$ g = \frac{1}{|\...
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For a proper cone $K$, $x \in \operatorname{int}(K), y \in K$, implies $x+y \in \operatorname{int}(K)$?
Let $K$ be a proper cone in $\mathbb{R}^{n}$, which is a cone that is convex, closed, pointed, and has a nonempty interior. If $x \in \operatorname{int}(K)$, and $y \in K$, must $x+y \in \operatorname{...
- 1,106
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Uniquness and relation between bi-invariant metric and bi-invariant distance?
During my research I came across Riemannian geometry. Specificlaly I am dealing with $SO(3)$.
I searched on the internet and there are multiple references telling me that a simple, compact Lie group ...
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Why is this subset associated to a $2$-tensor open and dense?
Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the ...
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Sum of derivatives of christoffels, vector laplacian on basis
One can define the connection laplacian on $TM$ as
$$
\Delta X := \sum_i \nabla_i \nabla_i X - \nabla_{\nabla_i \partial_i} X
$$
where $\{\partial_i\}$ give an orthonormal basis at a point, induce by ...
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