All Questions
103 questions
7
votes
2
answers
396
views
Is every metric uniformly close to a metric with negative scalar curvature?
Let $M$ be a smooth manifold with non-empty boundary.
Let $g$ be a smooth Riemannian metric on $M$. Is the following true?
For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ ...
- 6,741
1
vote
0
answers
92
views
Natural measures of curvature of Riemannian manifold along two-dimensional subspace
Given a Riemannian manifold $M$, a point $p \in M$, and some two-dimensional subspace $\varSigma$ of $T_{p}M$, the sectional curvature $K(\varSigma)$ is a well-known, natural measure of the curvature ...
- 965
9
votes
0
answers
283
views
Hermitian sectional curvature
Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.
...
- 1,347
6
votes
0
answers
201
views
Formula for difference between curvature operators?
This is a re-editing of a prerviously posted question:
Let $(M,g)$ be a Riemannian manifold. Let $C:TM\to TM$ be symmetric positive definite. Define the metric
$$
(X,Y)_C = (X,CY)_g.
$$
Denote by $\...
- 797
6
votes
1
answer
559
views
Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature
Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
- 123
2
votes
0
answers
216
views
A geometric rank of Riemannian manifolds
There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...
- 356
3
votes
1
answer
111
views
Sectional curvatures under simple maps
Suppose that we have a submanifold $X$ of $\mathbb{R}^n$ with the induced Euclidean metric, whose sectional curvatures we have a handle of (say, they are lower bounded by some $\kappa$).
Is there a ...
- 265
6
votes
1
answer
508
views
Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)
It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...
- 265
4
votes
1
answer
308
views
Realizing the cross product of $\mathbb{R}^3$ as the curvature tensor of a Riemannian metric on $\mathbb{R}^3$
Is there a Riemannian metric on $\mathbb{R}^3$ for which the corresponding curvature tensor $R$ satisfies $R(X,Y)Z=(X\wedge Y)\wedge Z$?
I have already discussed this question in the following post ...
- 356
3
votes
1
answer
174
views
Flatness in a neighborhood of a point condition
Suppose that we have a Riemannian Manifold $(M,g)$ whose
curvature vanishes in an open neighborhood U of a point p.
When does this imply that the metric is Flat ?
In particular, does it happen ...
- 303
2
votes
2
answers
515
views
A kind of "Curvature tensor" for higher dimensional tensors
I begin my question with a multilinear question then I will consider two local smooth analogies:
Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\...
- 356
4
votes
1
answer
303
views
Locally Riemannian Connection
Let $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ be a symmetric connection whose curvature is $$R^a{}_{bcd}=\partial_c\Gamma^a{}_{bd}-\partial_d\Gamma^a{}_{bc}+\Gamma^a{}_{ec}\Gamma^e{}_{bd}-\Gamma^a{}_{ed}\...
2
votes
1
answer
2k
views
Second fundamental form and embeddings
Let $\Sigma$ be a smooth hypersurface of a $d$ dimensional smooth Riemannian manifold $(\mathcal M, G)$;
we may see $G_x$ as a mapping from $T_x(\mathcal M)$ into $T_x^*(\mathcal M)$ so that
$$
\...
- 16.2k
12
votes
1
answer
454
views
Riemannian vs Non-Riemannian curvature
If you neither know the metric nor the holonomy group, how do you recognize a curvature tensor is Riemannian?
I assume a curvature, by definition, satisfies Bianchi identities. I know it is ...
1
vote
1
answer
550
views
Under what condition the covariant derivative of Ricci operator along Killing vector field vanish?
Let $(M,g)$ be a Riemannian manifold and $V$ a unit Killing vector field on it. Under what condition on curvature tensor the following equation hold:
$$\nabla_VQ=0,$$
where $Q$ is the Ricci operator ...
- 4,195
3
votes
0
answers
165
views
Flat Riemannian metrics adapted to quadratic vector fields with center
Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$
Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
- 356
4
votes
0
answers
495
views
Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)
This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.
Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
- 356
7
votes
1
answer
2k
views
Reference for parallel transport around loop and its relation to curvature
It is a well known fact that the geometric meaning of a linear connection's curvature can be realized as the measure of a change in a fiber element as it is parallel transported along a closed loop.
...
- 2,792
1
vote
2
answers
484
views
Derivation of the volume preserving mean curvature flow
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Picture above is from
Huisken, Gerhard, The volume preserving ...
- 165
2
votes
1
answer
837
views
A kind of surfaces
Is there a way* to prove that a Bonnet Surface $S$ in isothermal coordinates in $R^3$ with mean curvature ($H$) and Gaussian curvature ($K$) both non-constant and where $(H^2-K)=c$, (with c positive ...
- 272
0
votes
0
answers
571
views
Surfaces in isothermal coordinates and particular PDE
From Brioschi's formula, if we have a surface in isothermal coordinates were $g_{ij}=E(u,v)*\delta_{ij}$ is the metric tensor, the gaussian curvature is:
$K=-\frac{1}{2E}[\frac{\partial}{\partial u}(...
- 272
3
votes
0
answers
101
views
Conformal Transformations that are Ricci Positive Invariant
Is there any known class of conformal transformations $\phi : M \to M$ of a riemannian/semi-riemanian manifold $(M,g)$ that have the property: $g$ is ricci-positive iff $\phi^* g$ ricci positive?
...
- 31
5
votes
1
answer
209
views
Evolution of $W_+$ and $W_-$ under the Ricci flow
In dimension $4$ the Weyl operator $W$ splits in two parts
$$W_+:\Lambda^{2}_{+} \to \Lambda^{2}_{+}$$
and
$$W_-:\Lambda^{2}_{-} \to \Lambda^{2}_{-}.$$
(a) Has there been a study of the evolution ...
2
votes
3
answers
593
views
Curvature of singular Riemannian metric
Let $M$ be a differentiable manifold of dimension $n>1$ and $g$ a flat Riemannian metric on $M$. Consider $f:M\rightarrow \mathbb{R}^+$ a continuous function which doesn't have continuous first ...
user47069
7
votes
1
answer
558
views
minimal surfaces in $S^n$
Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...
- 914
17
votes
2
answers
1k
views
Are there some intrinsic invariants of surfaces other than Gaussian curvature?
The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
- 356
6
votes
0
answers
270
views
On the curvature tensor with certain conditions
Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace $ be a locally orthonormal frame for $M$($3\leq n,m$).
If we suppose the curvature tensor $R$ of $g$ ...
- 601
4
votes
0
answers
214
views
Flows associated with Killing fields
Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
- 1,378
1
vote
1
answer
321
views
Soliton equation and non-killing potential vector field
I am searching for a non-Killing vector field $\zeta \in\frak X\rm (M)$ where $(M,g)$ is a Riemannian manifold such that
$$\frac12 \frak L_\zeta \rm g+Ric=\lambda g$$
$$ \frak L_\zeta \rm Ric=\lambda \...
- 422
5
votes
1
answer
284
views
Compact Eucledean hypersurfaces with "almost" constant H_k curvature
Let $M$ be an Eucledean $n$-dimensional compact hypersurface with constant $H_k$ curvature, where $k=1,...n$. A theorem by A.Ros tell us that so $M$ is an Eucledean sphere. Does anybody know if there ...
13
votes
2
answers
1k
views
What is known about Lie groups with (strictly) positive curvature?
If we consider $G$ a compact Lie group, there is a left invariant Riemannian metric whose the sectional curvature is nonnegative (see Milnors' paper). When can we find a left invariant metric that has ...
- 245
3
votes
0
answers
367
views
Obtaining the metric from the mixed Ricci tensor $R^i{}_j$
In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$).
But what do we know about ...
- 4,312
3
votes
1
answer
560
views
Prescribing an induced metric
We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial f(x,y)}{\...
- 267
4
votes
1
answer
747
views
The Laplacian of an expression involving the Ricci tensor
While doing some computations on a compact Riemannian manifold I have reached the following expression:
$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$
where $\Delta_y$ is the ...
- 5,407
17
votes
2
answers
5k
views
Square of the distance function on a Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function
$$dist^2\colon M\times M\to \mathbb{R}$$
given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
- 21.8k
4
votes
0
answers
128
views
Normal fields of geodesic spheres
This question is related to this one (https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let $(M,g)$ be a compact Riemannian ...
- 345
2
votes
1
answer
664
views
Limited expansion of mean curvature of geodesic spheres
I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes ...
- 5,407
4
votes
1
answer
412
views
Symmetries of non-Riemannian curvature tensor
The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...
- 690
2
votes
0
answers
136
views
Bounding distance between geodesics in manifolds with nonpositive curvature
This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ...
- 191
2
votes
0
answers
354
views
Spherical cap is the only compact constant mean curvature surface bounded by a circle
I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap.
This is stated in the ...
5
votes
1
answer
291
views
Gaussian Curvature of Exponentiated 2-Planes
Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...
- 1,378
7
votes
5
answers
4k
views
Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?
Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat?
I am especially interset in the case ...
- 977
2
votes
1
answer
1k
views
curvature and volume growth
Let $M$ be a non-compact connected Riemannian manifold with $\mathrm{sec}_g=0$ and $\operatorname{vol} B(x,r)\geq c(n)r^n$ for any $r$, where $c(n)>0$. How to prove that $(M,g)$ is isometric to $(R^...
- 21
0
votes
1
answer
257
views
volume of a submanifold implies bounds on curvature
I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred ...
6
votes
0
answers
269
views
Negative curvature in the middle of $R^{3}$
What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...
- 61
7
votes
1
answer
1k
views
About Sectional Curvature [closed]
In a paper by Yann Ollivier:
Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint
of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
- 1,303
3
votes
1
answer
583
views
geometric meaning of Ricci-flatness
What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
- 1,303
26
votes
5
answers
7k
views
Intuition for mean curvature
I would like to get some intuitive feeling for the mean curvature. The mean curvature of a hypersurface in a Riemannian manifold by definition is the trace of the second fundamental form.
Is there ...
- 693
3
votes
1
answer
5k
views
Relationship between sectional curvature, bisectional curvature and conjugate points
Tian has defined bisectional curvature for unit and perpendicular tangent vectors $X,Y$ as follow
$$R(X,Y,X,Y)+R(X,JY,X,JY).$$
If bisectional curvature be constant, is there any relationship between ...
- 105
46
votes
2
answers
11k
views
Does the curvature determine the metric?
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that they are
not ...
- 1,755