All Questions
103 questions
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
36
votes
10
answers
8k
views
Some questions about scalar curvature
Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:
...
- 29.2k
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
20
votes
3
answers
9k
views
Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
- 251
18
votes
1
answer
1k
views
Is the minimal volume a topological invariant?
On Wikipedia, it is said that the minimal volume
$$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$$
is a topological invariant, ...
- 609
18
votes
1
answer
1k
views
Geometric interpretation of the Weyl tensor?
The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops.
Question: Is there a similarly direct geometric ...
- 63.9k
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
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
14
votes
4
answers
6k
views
When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?
I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
- 54.6k
14
votes
1
answer
1k
views
Progress on Gromov's Conjecture of the bound of total Betti numbers
This question is a reference request.
Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
- 175
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
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 ...
12
votes
0
answers
262
views
Jacobi fields on non-geodesic curves
The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for ...
- 1,277
10
votes
1
answer
3k
views
Taylor expansion of the metric tensor in the normal coordinates
I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates.
The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},...
- 45k
9
votes
1
answer
344
views
Do geodesics avoid regions where the curvature diverges?
Let $(M^2,g)$ be a Riemannian manifold, with manifold boundary $\partial M$. We assume that the metric degenerates at the boundary, in the sense that the (Gauss) curvature diverges like $K \to +\infty$...
- 5,038
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
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
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
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
7
votes
1
answer
197
views
Positively curved manifold with collapsing unit balls
Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and
$$
\inf_{p \in M} \text{Vol}_gB(p,1)=0?
$$
- 2,535
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
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
7
votes
0
answers
115
views
The space of positive scalar curvature metrics on $S^4$
Let $\mathcal{R}_{+\mathrm{sc}}(S^n)$ denote the space of complete Riemannian metrics of positive scalar curvature on the sphere $S^n$. It's known that $\mathcal{R}_{+\mathrm{sc}}(S^2)$ is ...
- 5,596
7
votes
0
answers
1k
views
Conventions for Riemann curvature tensor
I am aware of two conventions for the Riemann curvature tensor, namely the expression
$$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$
is either declared to be $R(X,Y,Z,W)$ or $...
- 18.7k
6
votes
2
answers
753
views
Curvature of nonsymmetric metric tensors?
Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...
- 327
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
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-...
- 9,237
6
votes
1
answer
378
views
Unbounded sectional curvature implies infinite diameter?
Let $(M,g)$ be a Riemannian manifold such that for each $C>0$ there is $p\in M$ and $X,Y\in T_pM$ unitary such that $K(X,Y) > C.$ Does this imply that the diameter of $(M,g)$ is infinite?
I ...
- 1,774
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
6
votes
3
answers
368
views
Curvature function as a random variable with uniform distribution
Let $(S,g)$ be a Riemannian surface. Then the curvature function $\kappa: S\to \mathbb{R}$ can be counted as a random variable. So it produces a probability density function
$f_g:\mathbb{R}\to \...
- 356
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
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
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
5
votes
2
answers
340
views
Example of a curvature with no associated metric
Is there a concrete example of a $4$ tensor $R_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e.
\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} =...
- 113
5
votes
1
answer
343
views
Clarifying a result of Klingenberg
I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
- 163
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 ...
5
votes
2
answers
379
views
Correct curvature tensor of symmetric space of positive definite matrices with trace metric?
Let $Pos(n)$ be the set of $n \times n$ real positive definite matrices with trace (aka affine-invariant) metric
$$\langle u, v \rangle_p = tr(p^{-1} u p^{-1} v)$$
for all $p \in Pos(n)$ and $u, v \in ...
- 637
5
votes
1
answer
156
views
Positively curved metric with uniformly positive scalar curvature
Can we find a complete noncompact Riemannian manifold $(M^n,g)$ with bounded geometry satisfying the following conditions?
the curvature operator $Rm>0$;
the scalar curvature $R \ge 1$.
Notice ...
- 2,535
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 ...
5
votes
1
answer
245
views
Possible sign of scalar curvature for Einstein warped product manifold with Ricci-flat
Let $(M, g_M)$ where $M= B \times_f F$ and $g_M=g_B + f^2g_F$, an Einstein warped product manifold (i.e., $Ric_M= \lambda g_M$), with Ricci flat fiber-manifold $F$, i.e., $Ric_F=0$.
Then $M$ can admit ...
- 272
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
5
votes
0
answers
445
views
Upper bound on the sectional curvature of a Riemannian submersion
Consider the manifold $M := \operatorname{SO}(n) \times \mathbb{S}^{n-1}$, endowed with the product metric given by the bi-invariant metric of $\operatorname{SO}(n)$ and the round metric of $\mathbb{S}...
- 45
5
votes
0
answers
244
views
Möbius strip zero curvature [closed]
Is there a Möbius strip, seen as an embedded surface in $\mathbb{R}^3$, with zero curvature? I know one can see the Möbius strip as the quotient of the square with reverse identification of two sides ...
5
votes
0
answers
101
views
How is this product of tensors defined?
I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...
- 2,095
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}\...
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
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
4
votes
1
answer
439
views
Etymology “Kulkarni–Nomizu product”
$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...
- 1,429
4
votes
1
answer
245
views
Tzitzeica surface
A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface ...
- 272
4
votes
1
answer
1k
views
Relation between mean curvature and conformal metric
We'll consider $(N, g)$ a Riamannian Manifold and $\overline{g} = e^{2f}g$ a conformal metric. Let M be a hypersurface in N, $\overline{H}_M$ and $H_M$ the mean curvature of M with respect
to the ...
- 55