Questions tagged [riemannian-geometry]
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
3,083 questions
8
votes
1
answer
218
views
Existence of properly discontinuous and cocompact action
Let $M$ be a complete Riemannian manifold. My question is, under what conditions does $M$ admit a discrete group of isometries $\Gamma$ which acts properly discontinuously and cocompactly on $M$, that ...
- 25k
0
votes
0
answers
246
views
A question about second fundamental form of Riemannian isometric embedding
I have got a question unsolved for some time. I do not know whether it is trivial or not:
**I omit a very important fact: The metric at point p is second-order flat, i.e. $d_p \phi(-,v) = 0$ and $d_p^...
- 21
36
votes
10
answers
6k
views
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 4,713
5
votes
1
answer
371
views
Lower bound on the first eigenvalue of the Lichnerowicz Laplacian on positive Einstein manifolds
Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless ...
CommunityBot
- 1
3
votes
0
answers
127
views
Is the normalized Ricci flow real analytic in the time variable?
Let $(M^n,g)$ be a closed Riemannian manifold. In this paper, B. Kotschwar proved that the Ricci flow $g(t)$ with initial condition $g(0) = g$ is real analytic with respect to the time variable, for $...
- 2,095
1
vote
1
answer
218
views
Fixed locus of a Kahler $S^1$-action
Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that
Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold.
It splits $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
- 1,677
10
votes
2
answers
862
views
Deforming metrics from non-negative to positive Ricci curvature
Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one?
I know it's impossible in general due to ...
- 4,195
8
votes
0
answers
409
views
What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?
In his paper [2], Paul Ehrlich write
In [1], Aubin stated a theorem which implied as a corollary that if a manifold
$M$ admits a Riemannian metric with nonnegative Ricci curvature and
all Ricci ...
- 4,195
3
votes
1
answer
183
views
Flat manifolds are local geometric objects
In an article called synthetic geometry in Riemannian manifolds M. Gromov says that
tori (and in general flat manifolds) must be seen as local geometric objects.
He does so after making an example ...
- 223
5
votes
0
answers
126
views
Metric under Ricci flow on a 2-sphere can be realized by embedding
I am sorry if this is a silly question, but I am new to Ricci flows.
Let $\Sigma \subset \mathbb{R}^3$ be a smoothly embedded sphere, and denote its metric (induced by $\mathbb{R}^3$) by $g$. Suppose ...
- 2,095
13
votes
3
answers
6k
views
Linear/Non-linear sigma model
This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model.
I gauge from some other interactions that it has considerable ...
- 2,821
2
votes
1
answer
119
views
Density of smooth bi-Lipschitz maps in smooth maps
Setup/Motivation:
Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class ...
- 26.3k
2
votes
0
answers
265
views
Riemannian geometry of Grassmannian bundles
The Grassmannian bundle of a vector bundle $E$ is a smooth manifold where each fiber over the base space is replaced by the Grassmannian (of specified rank) of the fiber. I am interested in defining a ...
- 23.3k
4
votes
1
answer
346
views
Yamabe operator, conformal transformations and square of the Dirac operator
On an $(M,g)$ compact n-dimensional Riemannian manifold the Yamabe operator $Y = 4(n-1)/(n-2)L + s$, where $L$ is the Laplacian and $s$ is the scalar curvature is conformally covariant, which for me ...
- 1,713
1
vote
0
answers
157
views
The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
1
vote
0
answers
83
views
Right-invariant metrics on the unitary groups and embeddings in the complexification
Let $G = SU(n)$ and $G_c = SL(n, \mathbb{C})$. Let $g$ be a right-invariant metric on $G$ and let $g_k$ be the Killing metric on $G_c$.
Define the map $p$ from $G_c$ to $G$ which maps $h \in G_c$ to
$$...
- 5,215
5
votes
0
answers
2k
views
Relationship between geodesics and curvature lines on surfaces?
I'm trying to understand the relationship between geodesics and lines of principal curvature (to keep things simple, let's say Riemannian 2-manifolds embedded in $\mathbb{R}^3$). In my reading, I ...
- 1,446
5
votes
1
answer
650
views
The vanishing of covariant derivative of an alternative metric tensor
Let $(M,g)$ be a Riemannian manifold, endowed with the Levi-Civita connexion $\nabla$ induced by $g$. By the very definition of the Levi-Civita connexion $\nabla$, we indeed know that $\nabla g=0$, i....
CommunityBot
- 1
2
votes
0
answers
268
views
Geometric characterizations of conformal maps
I have some fairly basic questions with regards to conformal structures/maps -- I apologize if they are on the basic side for MO, but I figured I might get some clarity here.
Suppose $X$ and $Y$ are ...
- 1,075
1
vote
0
answers
238
views
Example of CAT($k$) space [closed]
Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...
- 103
6
votes
1
answer
463
views
Holonomy bounded in terms of area and the curvature
I suppose the following result follows
from Ambrose-Singer theorem, but I cannot
find a reference, and the arguments I found
in the literature are usually weaker. The idea
is that holonomy over a null-...
- 271
3
votes
1
answer
486
views
There exists differentiable curves arbitrarily close to the continuous ones
Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma(t))$.
...
- 2,806
3
votes
0
answers
67
views
Combinatorial approximation to the integral of a form?
This is a bit of a followup to my previous question Intuition for the volume form - combinatorial definition?. I am looking for a certain combinatorial intuition when it comes to integrating ...
- 1,075
19
votes
1
answer
819
views
All saddles in the unit ball have area $<2\pi$?
Let $M$ be the saddle surface in $\mathbb R^3$ defined by $x^2-y^2+z=0$. For any $r\geq 0$ and $(x_0,y_0,z_0)\in\mathbb R^3$, let $rM+(x_0,y_0,z_0)$ denotes the surface obtained by scaling $M$ by $r$ ...
- 61.9k
0
votes
0
answers
321
views
Why are holomorphic $p$-forms parallel?
Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection.
It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e....
- 175
8
votes
0
answers
295
views
Intuition for the volume form - combinatorial definition?
I apologize that this is short of research level but I have realized that I am not happy with my understanding of the volume form on an oriented Riemannian manifold and I was hoping to find some ...
- 1,075
3
votes
0
answers
130
views
Is the range of the exterior covariant derivative closed in $L^{2}$?
Let $(M,g)$ be a compact Riemannian manifold. Given a tensor bundle $\mathbb{E}$, let $\nabla:\Gamma(\mathbb{E}) \rightarrow \Gamma(T^{*}M\otimes \mathbb{E})$ be the canonical connection induced by ...
- 808
2
votes
0
answers
120
views
Using Rauch comparison theorem to get a comparison of two metric
Picture below is from Topping's Lectures on the Ricci flow. I've been stuck by the red line about two months. In fact, I asked it on ME two months ago. To describe the problem more precisely, ...
- 165
2
votes
1
answer
292
views
The relationship about sub-unit ball and sub-elliptic ball
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n}$ with smooth boundary.$\{X_{1},\cdots,X_{m}\}$ be smooth real vector fields on $\Omega$ Which satisfy the Hormander condition. If $\gamma$ is ...
CommunityBot
- 1
5
votes
1
answer
327
views
Can we prescribe the $L^2$ norm of the scalar curvature on a four-manifold?
As mentioned by Willie Wong, I modified to the following version:
Let $M$ be a closed smooth $4$ manifold.
Q
Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, ...
- 19.3k
6
votes
2
answers
435
views
The convex hull of a manifold whose cobordism class is trivial
Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...
- 6,367
1
vote
0
answers
82
views
Exponential map on the Hilbert manifold $\Omega(M,q_0,q_1)$
Consider $M$ to be a compact manifold and consider the based loop space $\Omega(M,q_0,q_1)$ of loops of class $W^{1,2}$. It can be shown that that this has a structure of an hilbert manifold. It's ...
- 791
5
votes
0
answers
219
views
Is the volume functional analytic in the space of embeddings? What about locally?
Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...
- 2,095
3
votes
0
answers
117
views
Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow
In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
- 659
3
votes
1
answer
82
views
Closed almost geodesics in a Riemannian manifold
Let $M$ be a smooth Riemanniann manifold. For $\varepsilon \geq 0$ we call an $\varepsilon$-geodesic (I am not sure that this is a standard name) a smooth map
$$\gamma\colon [a,b]\to M$$
such that for ...
- 10.6k
23
votes
2
answers
6k
views
Why are they called isothermal coordinates?
On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:
$$g_{ij} = e^{f} \delta_{ij}$$
My question is:...
4
votes
0
answers
68
views
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 ...
- 518
0
votes
1
answer
169
views
Explicit computation of the vertical and horizontal vector bundles
Given a closed Riemannian manifold $(X,g)$ and let $p\colon TX\to X$ be the usual projection, the paper I'm reading asserts that the Levi-Civita connection induces a splitting $T(TX)= H(TX)\oplus V(TX)...
- 247
1
vote
1
answer
76
views
Cross product of two infinitesimal bendings
Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$.
Let $Z_1,Z_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential ...
- 89
3
votes
0
answers
187
views
Exponential map of cotangent bundle and Morse theory on based loop space
Consider $M$ to be a compact manifold and $q_0,q_1\in M$. Let $L_t$ be a lagrangian and $\mathcal{E}_L$ the lagrangian action functional on the based loop space $\Omega(M,q_0,q_1)$ defined as $\...
- 791
11
votes
1
answer
550
views
Does the first Laplacian eigenfunction on a homogeneous space have a unique maximum?
For convex domains $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, it's known that any first Laplacian eigenfunction is log-concave. In particular, it has a unique maximum.
These are ...
- 108k
1
vote
0
answers
128
views
Self-ajointness of the Laplacian over a Riemannian manifold with boundary
I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf).
Let
$(M,g)$ be a Riemannian manifold with boundary;
$E\to M$ be an hermitian fiber bundle;
$\Delta$ ...
- 141
2
votes
1
answer
114
views
Convergence of extremal subsets in Alexandrov spaces
Let $\{X_i^n\}$ be a sequence of $n$-dimensional Alexandrov spaces with curvature uniformly bounded from below which converges in the Gromov-Hausdorff sense to a compact $n$-dimensional Alexandrov ...
- 29.1k
2
votes
0
answers
185
views
Norm of a $(1, 1)$ form on a Kähler manifold
Given a Kähler manifold $(M, g)$ what is the convention for defining the inner product on two $(1,1)$ forms $\alpha = \sqrt{-1}\alpha_{i \bar k} dz_{i} \wedge d \bar{z}_{k}$ and $\beta = \sqrt{-1}\...
- 153
3
votes
0
answers
71
views
Prescribing variations that preserve the area
Let $(M^3,g)$ be a Riemannian manifold and let $\varphi : \Sigma \to M$ be a two-sided embedding of a closed surface into $M$, with a unit normal $N$. Suppose that $\varphi$ is a regular point of the ...
- 2,095
2
votes
0
answers
89
views
Manifold with totally geodesic boundary is spin if and only if its double is spin
Let $(M,g)$ be a Riemannian manifold with totally geodesic boundary $\partial M$. Let $(DM,Dg)$ be the double of $(M,g)$ obtained by reflection of across $\partial M$. I'm looking for a reference for ...
- 308
1
vote
1
answer
288
views
A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used
I'm reading Tawfik - The Yamabe problem: the PDE is
$$
\Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{q-1}. \label{1}\tag{1}
$$
Theorem (Yamabe). For $2<q<N=N=2 n /(n-2)$, there exists a $C^{\...
- 39.1k
2
votes
0
answers
65
views
Connection between a function and its usage in geometry [closed]
I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...
- 121
3
votes
0
answers
107
views
Changing the system of PDE by diffeomorphism
This problem comes from the book Hamilton's Ricci flow.
Given a smooth functional $f$, and following system. $$\partial_tg_{ij}=-2(R_{ij}+\nabla_i\nabla_jf)$$ If there exist a 1 parameter family of ...
- 273
6
votes
1
answer
299
views
Weitzenböck formula and comparison of norms
Let $M$ be a closed Riemannian manifold with a spin$^\mathbb{C}$ bundle $S$. Now for a spin connection $A,$ and a spinor $\phi,$ it can be shown that $C\lvert\nabla_A\phi\rvert^2\geq \lvert D_A\phi\...
- 808