All Questions
103 questions
2
votes
0
answers
72
views
Diameter bounds by mean curvature and area
I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$,
$$\text{diam}(\...
- 337
1
vote
0
answers
122
views
Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds
Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
- 111
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
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
0
votes
1
answer
74
views
Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems
I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries.
In pseudo-Riemannian geometry, for ...
- 45
2
votes
0
answers
123
views
Asymptotics on the number of diffeomorphism classes in the Cheeger finiteness theorem
A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which ...
- 3,212
4
votes
0
answers
166
views
Parallel transport of global sections and Riemannian curvature
A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days.
Consider a (real) smooth ...
- 3,513
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
3
votes
0
answers
102
views
Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields
Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection.
...
- 356
3
votes
0
answers
165
views
A higher-dimensional "line of curvature"?
Let $M$ be a submanifold of a Riemannian manifold $Q$, and let $\Gamma$ be a $d$-dimensional submanifold of $M$, i.e., $\Gamma^{d} \subset M \subset Q$.
Suppose that, for all (unit) normal vectors of $...
- 985
0
votes
1
answer
100
views
Curvature tensor of interpolation of two metrics
Let $\hat{g}$ and $\bar{g}$ be two smooth Riemannian metrics defined, say, on $\mathbb{R}^n.$ Consider a smooth function $\xi$ that acts as an interpolation function between the two metrics above on ...
- 85
1
vote
0
answers
210
views
Ricci-flat metrics on complex tori of dimension $n \geq 3$
Let $\mathbb{T}^n = \mathbb{C}^n /\Lambda$ be a complex torus of (complex) dimension $n$. If $n=2$, it is a theorem of Berger that the Ricci-flat metrics on $\mathbb{T}^2$ are flat. This follows from ...
- 1,379
0
votes
1
answer
117
views
Curvature of an affine system
I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It reminded me that many textbooks on Riemannian geometry only tell us about metrics, ...
- 111
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
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 ...
2
votes
0
answers
126
views
Conformal changes of metric and normal coordinates
Suppose that $(M,g)$ is a smooth Riemannian manifold of dimension $n\geq 2$. Let $p\in M^{\textrm{int}}$. Does there exist a small $\delta>0$ and a smooth function $c>0$ such that for the ...
- 4,145
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
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
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
1
vote
2
answers
284
views
Ricci scalar of submanifold of $\mathbf R^n$
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
where $\vec x$ are ...
- 521
1
vote
2
answers
148
views
Construct a hypersurface with fixed principal curvatures at a point
I'm reading Eschenburg's paper Local convexity and nonnegative curvature —
Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M$, $N\in T_pM$, we want to ...
- 241
0
votes
0
answers
252
views
Computation of scalar curvature from a Riemannian metric
I want to compute the scalar curvature for points on an empirical manifold (sampled data).
I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
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
2
votes
1
answer
224
views
The differentiability of the distance function on asymptotically flat manifolds
Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball.
Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard ...
- 969
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
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
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
2
votes
0
answers
101
views
Parallelism defect
I have a question that I don't know how to answer.
If I have a parallelism defect it is due to the presence of a curvature and therefore we can bring it back to a Riemann tensor.
The thing that is not ...
- 272
2
votes
0
answers
149
views
Comparison of sum of vectors and exponential map on a Riemannian manifold
Suppose $M$ is a simply-connected complete Riemannian manifold with bounded sectional curvature $\delta \leq K \leq \Delta < 0$. Let $p_0\in M$. Define the sequence of points $p_1, \ldots, p_n$ by
$...
- 41
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
4
votes
0
answers
148
views
The range of $\int_M \kappa_g ds$ where $g$ varies in all possible real analytic metrics on $M$
Let $M$ be a real analytic open surface(A non compact 2 dimensional manifold without boundary).
For every number $\lambda\in \mathbb{R}$, is there a real analytic Riemannian metric on $M$ with
$$\...
- 356
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
1
vote
1
answer
178
views
Asymptotics of constant mean curvature surfaces
Let $\Sigma^n \subset \mathbf{R}^{n+1}$ be a complete, properly embedded hypersurface with constant, non-zero mean curvature $H \neq 0.$
In the case where the dimension is $n = 2$, $\Sigma$ is non-...
- 5,038
1
vote
0
answers
97
views
Computing/estimating geodesics in practice
Let's say I have a Riemannian manifold with known metric $g$. I want to compute the geodesics of this manifold, specifically with respect to the Levi-Civita connection.
In practice, (i.e. with a ...
- 153
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
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
2
votes
2
answers
163
views
stability of two-sided sectional curvature bounds in Lorentzian geometry
Suppose that $(M,g)$ is a Lorentzian manifold of signature $(-,+,\ldots,+)$. Given a two plane $\Pi=\textrm{Span}\{X,Y\}$ with $X,Y \in T_pM$, we say that $\Pi$ is non-degenerate if
$$ g(X,X)g(Y,Y)-g(...
- 4,145
18
votes
1
answer
1k
views
Geometric interpretation of the Weyl tensor?
The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops.
Question: Is there a similarly direct geometric ...
- 63.9k
3
votes
1
answer
370
views
Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds
In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":
Suppose $(X,g_X)$ and $(Y,g_Y)$ are ...
- 392
0
votes
0
answers
126
views
mean curvature for codimension $>1$?
The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
- 3,625
3
votes
1
answer
551
views
Product formula for Laplace de-Rham operator
Let $M$ be a Riemannian manifold with Laplace de-Rham operator $\Delta = (d + \delta)^2$. If $g$ is a smooth $k$-form, and $f$ is a smooth function, is there a simple formula for $\Delta(fg)$ when $k &...
6
votes
2
answers
753
views
Curvature of nonsymmetric metric tensors?
Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...
- 327
12
votes
0
answers
262
views
Jacobi fields on non-geodesic curves
The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for ...
- 1,277
7
votes
1
answer
197
views
Positively curved manifold with collapsing unit balls
Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and
$$
\inf_{p \in M} \text{Vol}_gB(p,1)=0?
$$
- 2,535
5
votes
1
answer
156
views
Positively curved metric with uniformly positive scalar curvature
Can we find a complete noncompact Riemannian manifold $(M^n,g)$ with bounded geometry satisfying the following conditions?
the curvature operator $Rm>0$;
the scalar curvature $R \ge 1$.
Notice ...
- 2,535
4
votes
0
answers
880
views
Scalar curvature in terms of second fundamental form, reference request
I would like to cite a reference for the following formula for scalar curvature:
If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...
0
votes
0
answers
55
views
Gauss curvature of a fibre as a submanifold in a Riemannian warped product
Consider the Riemannian warped product $M^{n+1}=I\times\mathbb{S}^n$ with metric
\begin{align}
g=dt\otimes dt+f(t)^2g_{\mathbb{S}^n}
\end{align}
where $I\subseteq\mathbb{R}$ is some open interval and ...
6
votes
1
answer
378
views
Unbounded sectional curvature implies infinite diameter?
Let $(M,g)$ be a Riemannian manifold such that for each $C>0$ there is $p\in M$ and $X,Y\in T_pM$ unitary such that $K(X,Y) > C.$ Does this imply that the diameter of $(M,g)$ is infinite?
I ...
- 1,774
7
votes
0
answers
115
views
The space of positive scalar curvature metrics on $S^4$
Let $\mathcal{R}_{+\mathrm{sc}}(S^n)$ denote the space of complete Riemannian metrics of positive scalar curvature on the sphere $S^n$. It's known that $\mathcal{R}_{+\mathrm{sc}}(S^2)$ is ...
- 5,596
7
votes
0
answers
1k
views
Conventions for Riemann curvature tensor
I am aware of two conventions for the Riemann curvature tensor, namely the expression
$$\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