Questions tagged [riemannian-geometry]

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

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Definition of static spherically symmetric spacetime as fiber bundle

I am working on a physical paper about solutions of Einstein field equations in case of static spacetimes with perfect fluid spheres and wanted to invent a new definition of spherical symmetry there. ...
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Show the indefinite unitary group acts transitively on the hyperbloid manifold

Consider an indefinite Hermitian form $\langle \cdot ,\cdot \rangle$ on $\mathbb{C}^{n+1}$ such that $$\langle v ,w \rangle = \sum^n_{i=1} v_i \bar{w}_i - v_{n+1} \bar{w}_{n+1}. $$ We let $U(n,1)$ ...
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Calculating the product of the Riemannian manifolds and the Riemann curvature [closed]

I am a physics Master student currently taking a course on Riemannian geometry. In the course we are supposed to solve problem 7-4 out of Lee's Book Introduction to Riemannian Manifolds. The Problem ...
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1 answer
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How to calculate the volume of the image of the manifold

Let $M$ be a $n$ dimensional manifolds, $f:M \rightarrow \mathbb {R}^n$ be a smooth map. Then, how can I calculate $\textrm{vol}(fM)$ ? I'm thinking of calculating it using the area formula as shown ...
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What is the precise definition of the Darboux tangent to a surface? [closed]

What is the definition of Darboux tangents of a surfaces? The book "Affine Differential Geometry" mentions it, but it does not give a precise definition.
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2 answers
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An affine invariant notion of minimal surface?

The notion of minimal surface (i.e. having vanishing mean curvature) is not "affinely invariant" in the following sense: if $M\subset\Bbb R^n$ is an ($m$-dimensional) minimal surface, then $...
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Embedding $X \ni x \mapsto \delta_x \in P_2(X)$ is totally convex

I am looking for a reference to a proof of the following result: Let $X$ be a compact, connected, smooth Riemannian manifold. Then, the embedding $$X \ni x \mapsto \delta_x \in P_2(X) $$ has totally ...
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Isomorphism of pseudo-orthogonal group $O(p,q)$ and $O(p', q')$ for $p,q, p',q' \in N$ [closed]

Let $O(p,q)$ be the pseudo-orthogonal group or indefinite orthogonal group of signature $(p,q)$ as described in https://en.wikipedia.org/wiki/Indefinite_orthogonal_group. Suppose $O(p,q)$ is ...
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When does a Riemannian manifold admit conformally flat coordinates with a Killing coordinate?

I have found the upper half-space coordinates on hyperbolic space $\mathbb H^d$ highly technically convenient. In these coordinates the metric takes the form $$g = f(x)((dx^1)^2 + \cdots + (dx^{d - 1})...
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Why use orientation-preserving diffeomorphism (instead of all diffeo's) in the construction of the moduli space of a Riemannian manifold

The question is basically in the title, but I want to make it more precise: Given an oriented Riemannian 2-manifold $\Sigma$ one can take a quotient of the set $$ \mathcal{M}_+(\Sigma)=\{~c~\mid (\...
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Looking for a cuvature tensor that adds up linearly

I'm looking for a curvature tensor "$C$" that characterizes the curvature like the Riemann tensor but adds up linearly regarding the metric tensor: $C_{\mu\nu} (g_1+g_2) = C_{\mu\nu} (g_1)+...
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Properties of the linearized Riemann tensor?

The linearized Riemann tensor is given by: $$R_{\alpha \beta \mu \nu}=-\frac{1}{2}\left[h_{\alpha \mu, \beta \nu}+h_{\beta \nu, \alpha \mu}-h_{\alpha \nu, \beta \mu}-h_{\beta \mu, \alpha \nu}\right]$$ ...
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1 answer
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Unique representation of $\mathbf{Gr}^+(p,n)$ the oriented real Grassmannian

For $\mathbf{Gr}(p,n)$ the $p$ dimensional subspace of $\mathbb{R}^n$, or equivalently $O(n)/O(p)\times O(n-p)$, a point has a unique projector representation $P = UU’$ where U is an $n \times p$ ...
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Is the map $T_X |_S (p) := \exp{X(p)}$ a diffeomorphism onto its image?

Preliminaries The exponential map $\exp : TM \rightarrow M$ is defined by $\exp{(v)} = \gamma_v (1) $ where $\gamma_v$ denotes the geodesic starting at $p \in M$ and initial velocity $v \in T_p M$. ...
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A doubt on the proof of the isometric invariance of the Levi Civita connection

The Levi Civita connection is isometrically invariant: Note $F_*X$ denotes the pushforward of vector field $X$ Let $M$ and $M'$ be differentiable manifolds having riemmannian metric $g$ and $g'$. Let $...
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Lie derivative in Kahler manifold

I have the following question $X$- a compact Kahler manifold and $v\in \Gamma(X,TX)$- Killing vector field. I don't understand why Lie derivatives equal to zero $L_v \omega=0$ where $\omega$ is the ...
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2 votes
2 answers
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What is a homothetic change of metric? How does it mean that using a homothetic change of metric we can set the volume equal to 1.

I'm reading Aubin's work on Yamabe problem(the book Some Nonlinear Problems in Riemannian Geometry page150) He wrote that by a homothetic change of metric we can set the volume equal to one. So ...
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2 votes
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Continuity of the curve shortening process

I'm studying the shortening process, introduced in the book Course in Minimal Surfaces by T. Colding and W. Minicozzi, which is inspired by the Birkhoff's curve shortening process. In the book, is ...
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Convergence of minimal surfaces

I'm studying the proof of the Positive Mass Theorem given by Schoen-Yau ("On the Proof of the Positive Mass Conjecture in General Relativity"), but there's a detail I cannot understand. When ...
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Volume of geodesic ball in constant sectional curvature manifolds

$M$ is n-dimensional Riemannian manifold, and has constant sectional curvature $K_0$. When $r>0$ is small enough, denote $$ B(p,r) = \exp_p(B(r)) $$ where $B(r)\subset T_pM$ is a ball of radius $r$...
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is there a NURBS but have weights on different axis?

We know that NURBS has the form $p(u)=\frac{\sum_{i=0}^n\ w_i\ N_{i,k}\ (u)\bf d_i}{\sum_{i=0}^n\ w_i\ N_{i,k}\ (u)}$ which adds weights on different control point $\bf d_i$. I am looking for a form ...
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Question about the covering space of a spin manifold

I'm a little confused about how to prove that a covering space $\widetilde{M}$ of a spin manifold $M$ is also a spin manifold. If not, is there some counterexample? Could you please give me some help ...
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Question about the connected sum of two smooth manifolds

I'm a little confused with the following two questions about connected sum: (1) Is the covering space$\widehat{M\# N}$ of the connected sum of two smooth closed manifolds $M, N$ the connected sum $\...
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Conjugate points and expansion of the geodesic congruence

I am working in a Lorentzian manifold $(M, g)$ (but I think the problem would be quite similar in a Riemannian manifold) and I am considering a timelike geodesic whose tangent vector field is denoted ...
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2 votes
2 answers
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Calculating the gradient of Log-Euclidean distance between SPD matrices on Riemannian manifold

In the paper Log-Euclidean metrics for fast and simple calculus on diffusion tensors, the geodesic distance between SPD matrices $A,B$ is defined as $$d(A,B)=||\log A- \log B||_F,$$ where $F$ is the ...
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Approximation of a manifold at second order

The approximation of a manifold at first order at one point is the tangent space of the manifold at this point. Is there exist a notion of approximation at second order of a manifold at one point. We ...
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Show that $\varphi_A^*g=g$ (Isometries of the hyperbolic space)

Let $\mathbb H=\{(x_0,x_1,x_2) : B(x,x)=x_0^2+x_1^2-x_2^2=-1,x_2>0\}$ the hyperbolic space endowed with the metric $g=dx_0^2+dx_1^2-dx_2^2,$ and let $\varphi_A :\mathbb R^3 \to \mathbb R^3 : x\to ...
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Free boundary geodesics as a critical point of the energy functional

As a consequence of the formula for the first variation of the energy of a curve, we have the following known characterization of geodesics. A piecewise differentiable curve $c:[0,1]\to M$ is a ...
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What condition needs to be placed on a manifold for integration by parts to not have boundary term at infinity?

Say you have a complete, noncompact Riemannian manifold without boundary $(M,g)$. I am wondering what condition needs to be placed on the metric $g$ so that when you perform an integration by parts, ...
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Curvature and position of tangent space

Let $M$ be a hypersurface of $\mathbb{R}^d$, $x_0 \in M$ and $T$ the tangent space of $M$ at $x_0$. Which of the following are true ? If the sectional curvature of $M$ at $x_0$ is strictly positive ...
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1 vote
1 answer
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Kahler manifold computation

In the following derivation, done on a Kahler manifold, where $\nabla$ is the complexification of the Riemannian connection (i.e., since we are on a Kahler manifold, this is the same thing as the ...
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1 vote
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Lectures on differential forms and Riemannian Geometry

I've just taken an introductory course on Riemannian Geometry. I've read several chapters of Do Carmo's book Riemannian Geometry and O'Neill's one on Semi-Riemannian Geometry. Also, I am familiarized ...
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3 votes
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Developments of Riemannian geometry in recent decades.

I have very little knowledge about Ricci flow and Riemannian Geometry. I can't understand the evolution from the traditional Riemannian geometry to Ricci flow (or geometrical analysis ). Obviously, ...
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4 votes
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Simply connected in "An application of second variation to submanifold theory"

I read the section 3 of chapter 10 of do Carmo's Riemannian Geometry. In fact, it is reproduce of Moore, John Douglas, An application of second variation to submanifold theory, Duke Math. J. 42, 191-...
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0 votes
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Sobolev embedding implies lowerbound on the volumn of the ball

Assume Sobolev embedding: $$H_{1}^{1}(M) \subset L^{n /(n-1)}(M)$$ holds for the compact Riemannian manifold $(M^n,g)$, then we can get $$\operatorname{Vol}_{g}\left(B_{x}(r)\right)^{(n-1) / n} \leq C ...
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Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
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5 votes
1 answer
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Is this Lie group isometric to the Euclidean plane?

Consider the Lie group $$G := \left\{ \begin{pmatrix} 1 & 0 & 0 \\ x & 1 & 0 \\ y & x & 1 \end{pmatrix}: x, y \in \mathbb{R}\right\}$$ equipped with the left-invariant ...
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1 vote
0 answers
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Equivalent definitions of Sobolev space on manifold and references

It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$: D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm. D2. All functions ...
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Exponential map definition

I have a general question about the definition of the exponential map. I am taking the definition on page 72 of John M. Lee of "Introduction to Riemannian Manifolds" (https://www.maths.ed.ac....
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2 votes
2 answers
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Image of the exponential map on the 2-Sphere at a point

Take the north pole of the standard 2-sphere in $\mathbb{R^3}$ equipped with the riemannian metric induced by the standard metric on $ \mathbb{R^3}$. What is the image of the exponential map of the ...
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1 vote
1 answer
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Riemannian distance via the length of a curve

I am reading a book where they define the Riemannian distance between two points on a manifold. Naturally it is given as the infimum of the integral of the length of the curves which connect the two ...
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1 vote
1 answer
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Levi-Civita Connection with Geodesic Spray Containing Flow Lines of Time-Independent Vector Field?

Let $Q^n$ be a closed manifold and $M = TQ$ be its tangent bundle. In [1], it is worked out in Equation 5.31 that there is a kinetic energy Riemannian metric $g$ on $Q$ with Levi-Civita connection $\...
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2 votes
0 answers
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Compute Riemannian metric.

Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}^{3}$ given by $$ \varphi(x_{1}, x_{2}) = (x_{1}, 2x_{1}, x_{1}^{2} + x_{2}) $$ and consider the surface $S = \varphi(\mathbb{R}^{2})$, so $\varphi: \mathbb{...
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  • 337
3 votes
2 answers
74 views

Relating the Lie derivative to inner product of 2-tensors

Let $(M,g)$ be a Riemannian manifold. Let $f \in C^\infty(M)$, $X$ be a vector field and $h$ be a (symmetric) covariant $2$-tensor. Denote by $\langle \cdot, \cdot \rangle$ the inner product induced ...
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2 votes
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A coordinate free computation of the acceleration of $ \exp_p(t(v + \frac{t}{2}w))$

$\newcommand{\al}{\alpha}$ Let $M$ be a smooth Riemannian manifold. Fix $p \in M, v,w \in T_pM$, and define $\gamma(t) = \exp_p(t(v + \frac{t}{2}w))$. Is there a coordinate-free proof that $(\nabla_{\...
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Question about elliptic boundary condition

I'm a little confused with showing that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. Could give please give me some help with the details? Thanks in ...
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1 vote
1 answer
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Can every tangent vector be realized as an acceleration of a path with a given velocity?

$\newcommand{\al}{\alpha}$ Let $M$ be a smooth Riemannian manifold. Fix $p \in M, v \in T_pM$. Define $$ \mathcal{A}_v:=\{ w \in T_pM\,|\,\exists\alpha:(-\epsilon,\epsilon) \to M, \, \, \alpha(0)=p, \...
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1 vote
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Showing completeness - Hopf-Rinow Theorem

Let $g_0$ be the usual Euclidean metric on $\mathbb{R}^{2}$ and define Riemannian metric $g=(1+{x_1}^2+x_2^2)^{2}g_0$, also on $\mathbb{R}^{2}$. Show that if $\alpha:[0,L]\rightarrow \mathbb{R}^2$ is ...
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3 votes
2 answers
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In the proof of Lemma 6.18 associated to the Hopf-Rinow theorem.

I'm reading the John M.Lee, Introduction to Riemannian manifolds, second edition, p.167, Lemma 6.18 and I stuck at some statement : My question is, Question 1. Why can we write a unit-speed ...
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A basic question about calculation under geodesic frame

I have been recently reading some books related to Ricci flow, and I am confused when it comes to the evolution of Levi-Citita connection. That is: If $(M^n,g)$ Is a closed Riemannian manifold, and we ...
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