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Questions tagged [riemannian-geometry]

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

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Can the Shape Operator be defined on arbitrary Riemannian manifolds?

While reading Chapter 15 of Needham's Visual Differential Geometry (picture of the passage at the end of the post) I came to wonder a couple of things: What is the rigorous definition of the Shape ...
Sam's user avatar
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On writing integration on Riemannian submanifolds in terms of exponential map

I have a basic question about integrating (restriction of) functions to a Riemannian submanifold. Particularly, given a Riemannian manifold $M$, an embedded Riemannian submanifold $N \subseteq M$, and ...
Aniruddha Deshmukh's user avatar
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Integral over Curvature and metric components

Let $(M,g)$ a closed connected oriented manifold of dimension $n=2m$. Furthermore, let $\nabla$ be the Levi-Civita connection and $R$ is the Riemannian curvature tensor. Let $h$ be a symmetric $(0,2)$-...
SteuerWB's user avatar
-2 votes
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Fields related to differential geometry

I've been reading introductory texts in differential geometry, and I'm wondering what comes next. The most natural next step seems to be Riemannian geometry, but I also know that differential ...
DC2974's user avatar
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Riemannian Manifolds (Lee) Exercise 6-28 (d): Isometries converging pointwise converge topologically

Suppose $M$ is a connected, complete Riemannian manifold, $\mathrm{Iso}(M)$ is a smooth Lie group composed of all isometries of $M$, and $\phi_n$ is a sequence of isometries converging to an isometry $...
subrosar's user avatar
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Cheeger-Gromov limit of scalings of a noncompact Riemannian manifold

I am interested in the Cheeger-Gromov compactness theorem for sequences of Riemannian manifolds, and I stumbled across a question about smooth pointed Cheeger-Gromov convergence that I’m having ...
Jace167's user avatar
3 votes
1 answer
49 views

Prove simply-connected space forms satisfy SAS congruence axiom

We restate Hilbert's 6th axiom of congruence in the language of differential geometry: On a complete Riemannian manifold $M$, given geodesic triangles $[pqr]$ and $[p'q'r']$ with $\operatorname{dist}(...
Alkali Zeng's user avatar
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Generator of translations in a Riemannian manifold

Let $f$ be a smooth function from $X=\mathbb{R}^n$ into $\mathbb{R}$. We have that \begin{equation} e^{\vec{\tau}\cdot \vec{\nabla}}f(\vec{x})=f(\vec{x}-\vec{\tau}) \end{equation} where $\vec{\tau}\in ...
STU's user avatar
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Relationship between the spectra of the zero and first Hodge Laplacian on 2 dimensional manifolds.

Considering an oriented and compact surface embedded in $\mathbb{R}^3$. I would like to know if there exisits a particular relationship between the spectrum of the Laplace-Beltrami operator $\Delta_0 =...
Ricardo Gloria's user avatar
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In a geodesic triangle, is the longest side opposite to the largest angle?

If I have a complete (smooth) Riemannian manifold $(M,g)$ and three points on it, that I connect with distance minimizing geodesics, will the longest edge be opposite to the largest angle? In ...
PleaseAnswerMyQuestion's user avatar
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The representation has finite kernel.

Let $\rho: \Gamma \rightarrow \text{Isom} (\mathbb H^n)$ is a representation of finitely generated group $\Gamma$. Let $x_0 \in \mathbb H^n$ and the orbit map $\tau_\rho: \Gamma \rightarrow \...
yyffds's user avatar
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Homeomorphic Riemannian Manifolds are Isometrically [closed]

I had a question when I was learning Riemannian geometry: If two 2-dimensional Riemannian manifolds are homeomorphic, and for each pair of corresponding points, their sectional curvature is the same, ...
Tom of Halo City's user avatar
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Action of Isotropy group on hyperbolic spaces

Let $\mathbb R^{n,1}$ be the space $\mathbb R^{n+1}$ endowed with the bilinear form $$\langle x, y \rangle_{n,1} = \sum_{i = 1}^{n} x_iy_i -x_{n+1}y_{n+1}$$ We define the hyperboloid model as $$\...
yyffds's user avatar
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Some incorrect terms in generalizing Bochner's Formula

I am interested in Bochner's Formula but for slightly more general applications. In particular, I am interested in $\Delta g(U,V)$ where $U$ and $V$ are vector fields on a manifold. This involves ...
David Perrella's user avatar
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Geodesics on a Riemannian surface

I am studying Riemannian geometry and I am facing a problem with geodesics: if $S=(S,g)$ is a Riemannian surface, then $\gamma: I\rightarrow S$ is the parametrization of a geodesic if and only if, by ...
user1255055's user avatar
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Elliptic Equation on sphere

The literature I read recently says that the function $$\phi_1(\theta)=(\theta\cdot e_d)_+$$ defined on the sphere solves the equation $$ -\Delta _{\mathbb{S}}\phi _1=\left( d-1 \right) \phi _1\qquad \...
zik2019's user avatar
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Gaussian map and curvature for a sphere

I would kindly like to ask for your help regarding the Gauss map, I have studied that the Gauss map is the map $N: S \rightarrow S^2$ and has its differential: $dN_p: T_pS \rightarrow T_{N(p)}S^2 \...
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Gauss map and its differential

Good morning, I am looking for help to understand a concept related to the Gauss map $$N: M \rightarrow S^2$$ Since I am interested in the variation of $N$, I evaluate the differential and do not ...
user avatar
1 vote
1 answer
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Positive scalar curvature Einstein manifolds are noncollapsed

I am currently working through some of the exercises in Ricci Solitons in Low Dimensions by Bennett Chow, and I've been stuck on Exercise 1.23. It asks you to prove that positive scalar curvature ...
Will23's user avatar
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Existence of smooth triangulation for Riemannian 2-manifold

Most proofs that I can find of the Gauss-Bonnet Theorem for a compact Riemannian $2$-manifold $M$ always start with the assumption that $M$ has a smooth triangulation, i.e. a triangulation where the ...
Tob Ernack's user avatar
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Is squared geodesic distance about a point convex when defined on a convex ball centered at that point?

Let $\mathcal{M}$ be a complete Riemannian manifold. Let $x \in \mathcal{M}$, and let $r_x >0$ be the convexity radius at $x$. Let $B \subset \mathcal{M}$ be a geodesic ball centered at $x$ with ...
Spencer Kraisler's user avatar
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1 answer
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Isometry group of hyperbolic space

Let $\mathbb R^{n,1}$ be the space $\mathbb R^{n+1}$ endowed with the bilinear form $$\langle x, y \rangle_{n,1} = \sum_{i = 1}^{n} x_iy_i -x_{n+1}y_{n+1}$$ We define the hyperboloid model as $$\...
yyffds's user avatar
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Exploring the relationship between Tensorial Techniques and Coordinate-Independence?

I am studying Differential Geometry and Riemannian Geometry.I need some help from my stack exchange community members in this regard. Recently I happened to get a lecture series on Riemannian Geometry ...
Kishalay Sarkar's user avatar
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1 answer
53 views

The topology on Riemmanian manifolds as metric space [duplicate]

Let $M$ be a smooth manifold (no hypothesis on $M$ are done e.g. compactness or connectedness), then given $g_1$ and $g_2$ Riemmanian metric on $M$ we can see $M$ as a metric space with the distance:\...
Radagast's user avatar
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what is $\gamma'(t)$?

Here is the question I am trying to solve: $(a)$ Given $A \in PSL(2, \mathbb R),$ show that its derivative $D_z A$ satisfies $$g_{Az} (D_z Av, D_z Aw) = g_z(v,w)$$ for all $z \in \mathbb H^2$ and $u,w ...
Emptymind's user avatar
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Covariant derivative of a function along a curve

I am trying to find the covariant derivative of the function $f(x) = \arccos(\langle x, y\rangle)$ with respect to $x$ along a smooth curve using Christoffel symbols. My working space is $(S^{n},g)$ ...
vanilla-objective's user avatar
2 votes
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Upper bound on the sectional curvature of a Riemannian submersion

I already posted [this same question][1] on MathOverflow but received no answer so far so I am trying here just in case. Consider the manifold $M := \operatorname{SO}(n) \times \mathbb{S}^{n-1}$, ...
mathusername's user avatar
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Simplify $\nabla_{\nabla_{[W,X]}Y-\nabla_W\nabla_XY+\nabla_X\nabla_WY}Z[f]$

I'm trying to simplify the formula as in the title. I know that $$R(W,X,Y,Z)=\langle\nabla_{[W,X]}Y-\nabla_W\nabla_XY+\nabla_X\nabla_WY,Z\rangle$$ by the definition. Then I may conclude that $$\left(\...
一団和気's user avatar
1 vote
2 answers
85 views

The image of a Riemannian manifold under a distance-based mapping to $\mathbb{R}^n_{\geq 0}$

Consider $n$ points, $p_i\in M,\, i=1,\ldots , n$, in a Riemannian manifold $M$, and define the map \begin{array} &f: M&\to& \mathbb{R}^n_{\geq 0}\\ \qquad q &\mapsto& \left(\begin{...
STU's user avatar
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1 answer
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Identity for curvature tensor

I need to show the following idetnity, where $R$ is riemannian curvature tensor of a manifold $(M,g)$ and $x,y,z,t\in T_pM$: $$ \partial_\alpha\partial_\beta\{R(x+\alpha t,y+\beta z,y+\beta z,x+\alpha ...
Gao Minghao's user avatar
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1 answer
68 views

For a $q$-form $\varphi$, $\nabla\varphi=0$ if and only if $\nabla(*\varphi)=0$, where $*$ is the Hodge star operator.

I know the formula that connects differentiation and covariant differentiation is $$d \varphi\left(X_{1}, \cdots, X_{q+1}\right)=\sum_{i=1}^{q+1}(-1)^{i-1}\left(\nabla_{X_{i}} \varphi\right)\left(X_{1}...
HeroZhang001's user avatar
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1 vote
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Isometry group of quadric model of anti-de Sitter space

I am learning Lorentzian geometry on my own. Consider the space $\mathbb R^{p+2}$ endowed with the bilinear form $$\langle x, y \rangle_{p,2} = \sum_{i = 1}^{p+2} x_iy_i -x_{p+1}y_{p+1}-x_{p+2}y_{p+2}$...
yyffds's user avatar
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Derivative on a submanifold of $\mathbb{R}^n$ through a curve

I am a beginner in differential geometry and have been trying to find the gradient of the great arc distance $f(x) = d(x,y) = \arccos(\langle x, y\rangle)$ with respect to $x$, but getting two ...
vanilla-objective's user avatar
1 vote
2 answers
67 views

Details in the proof of a local version of Cartan-Ambrose-Hicks theorem

I am very sorry for the long length, but most of it is notions conventions and there isn't much argument and deduction. Let $ M^{n} $ and $ \bar{M}^{n} $ be two Riemannian manifolds. Fix base points $...
HeroZhang001's user avatar
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4 votes
1 answer
235 views

Why $\omega|_S\in\Omega^1(S,\mathfrak{g})$ iff $\omega(\nabla,e)\in\Omega^1(U,\mathfrak{g})$?

Let $M$ be a smooth manifold, let $\mathcal{S}$ be a $G$-strucutre on $M$ and let $\nabla$ be a connection on $TM$. Let $\omega\in\Omega^1(\text{Fr}(TM),\mathfrak{gl}_r)$ be the connection 1-form ...
Armando Patrizio's user avatar
1 vote
1 answer
59 views

Parallel vector field along a geodesic is a Jacobi field?

Let $M$ be a manifold with a linear connection $\nabla.$ A vector field $V$ along a curve $\gamma$ is said to be parallel along $\gamma$ with respect to $\nabla$ if $D_tV =0$. Let $\gamma : I \to M$ ...
Plantation's user avatar
  • 2,698
2 votes
0 answers
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Equivalence of Sobolev spaces Definitions on Riemannian manifold via $k$-Covariant Derivative and $k$-Gradient Derivative

I am studying Sobolev spaces on Riemannian manifolds and have encountered two different definitions in the literature. In the references: E. Hebey. Nonlinear analysis on manifolds: Sobolev spaces and ...
Raoní Cabral Ponciano's user avatar
2 votes
1 answer
60 views

Jet bundle question

Let $E \to M$ be a holomorphic vector bundle. Is there a metric on the first jet bundle $J^1 E$ that can be defined in terms of metrics on $E$ and $M$?
Hammerhead's user avatar
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Comparability Riemannian Metric in chart

Let $(M, g)$ be a Riemannian manifold. I know the following result: For each $x \in M$ there exists a chart $V$ and a constant $C \geq 1$ such that for all $y \in V$ and $\eta \in T_y \mathbb{R}^n$ $$ ...
Metalhead's user avatar
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25 views

Dual of the sum of dual metrics

Suppose I have a (smooth, connected, compact) Riemannian manifold $M$ and two smooth metric tensors $g_1$ and $g_2$. I can define the metric tensor $g:= (g_1^\star+g_2^\star)^\star$ where for a tensor ...
Bremen000's user avatar
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Vector field of rotation around an axis in spherical coordinates

This is a qualifying exam question in Differential Geometry. I'm new to the subject and am reading up from John Lee's Introduction to Smooth Manifolds. Let $M=S^2\subset\mathbb{R}^3$ be the unit ...
giraffe's user avatar
2 votes
1 answer
94 views

How can i prove the Darboux equation for a surface embedded into a 3D Riemannian Manifold?

I'm studying the generalizzation of the Darboux equation into immersions in Riemannian 3D space instead of 3D Euclidean space. Basically, in the Euclidean case, given a surface $\Sigma$ with metric $$...
Ghilele's user avatar
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2 votes
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Interpretation of Ricci curvature $Ric(V, W)$ when $V \neq W$

Let $Ric$ be the Ricci curvature tensor on a Riemannian manifold $M$. I am interested in the geometric interpretation of the Ricci curvature tensor. The explanation usually given is as follows. Fix $p ...
CBBAM's user avatar
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1 vote
1 answer
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Understanding the Relationship Between the Principal Symbol of $-\Delta$ and $\sqrt{-\Delta}$ and Geodesic Flow in Hamiltonian Systems

In the context of Hamiltonian systems in symplectic and Riemannian geometry, consider the following fact: Let $(M,g)$ be a Riemannian manifold and $(M,\omega,H)$ a Hamiltonian system with $$H(q,p)=\...
ayphyros's user avatar
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1 answer
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Existence of some point $ t' \in [0,b] \xrightarrow{\gamma} U_0 \subsetneq U$ ( unit speed radial geodesic ) satisfying certain property.

Let $(M,g)$ be a Riemannian $n$-manifold. $p\in M$. Let $U$ be a normal neighborhood of $p$, and $r$ be the radial distance function on $U$ ( C.f. Def. : John Lee's Introduction to Riemannian ...
Plantation's user avatar
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1 answer
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Fundamental equations of a Riemannian immersion

The Gauss, Codazzi-Mainardi and Ricci equations are the three fundamental equations of a Riemannian immersion. The Gauss equation inputs 4 tangent vectors, the Codazzi-Mainardi inputs 3 tangent ...
AlexInorbit's user avatar
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1 answer
72 views

Three simple(?) questions about proof of Lemma 10.18 in the John Lee's Riemannian manifold book.

I am reading the John Lee's Introduction to Riemannian manifold, Proof of the Lemma 10.18 and stuck at some statements : Lemma 10.18. Suppose $(M,g)$ is a Riemannian manifold with parallel curvature ...
Plantation's user avatar
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2 votes
0 answers
38 views

A curve in a single orbit in the space of Riemannian metrics

Let $M$ be a smooth manifold. Let $\mathcal M$ denote the collection of all Riemannian metrics on $M$. There is a right action of the product group $\mathbb R^+ \times \text{Diff}(M)$ on $\mathcal M$ ...
Joseph Kwong's user avatar
3 votes
1 answer
128 views

For every n-Riemannian manifold, is every local chart of nonzero radius isometrically embeddable in (n+1)-Euclidean space as Monge patch?

We hear common phrases such as "3-dimensional space curves into 4th dimensional flat space" being thrown around popular science media. And I want to confirm whether such statement is true. ...
Leon Kim's user avatar
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2 answers
66 views

Relation between Christoffel Symbols and metric tensor

I am learning about Riemannian geometry, and I have a question about relation between the Christoffel symbols and the metric tensor of the manifold. Is it true that the Christoffel symbols are unique ...
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