Questions tagged [curvature]
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
283
questions
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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 ...
0
votes
0
answers
138
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Kähler manifold with negative sectional curvature
Goldberg's theorem states that every almost Kähler manifold of constant curvature is Kähler if and only if the curvature is zero. This seems to contradict the fact that the sectional curvature of the ...
3
votes
1
answer
240
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Ricci flow and curvature
I am trying to read about geometric flows mainly Ricci flows. I have a question in mind, which I am not sure whether it's possible or not.
So my question is if one starts with a metric that has mostly ...
2
votes
0
answers
105
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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 ...
4
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0
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130
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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 ...
5
votes
1
answer
216
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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 ...
5
votes
1
answer
215
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Commutative/ symmetric second covariant derivative
Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$.
Is it possible to have an affine connection, possibly with non-zero ...
3
votes
1
answer
126
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Curvature calculation of $S^{2n+1}=U(n+1)/U(n)$ as a homogeneous space
I asked this question at StackExchange, but got no answer. So I am reposting it here.
I eventually want to check how the Hopf fibration
$$
S^{2n+1}\to {\mathbb C}P^n
$$
satisfies the Riemannian ...
0
votes
0
answers
41
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different definitions of holomorphic bisectional curvature
Peter Li and Jiaping Wang defined holomorphic bisectional curvature in their paper as follows:
Assume that $M^m$ is a Kahler manifold of complex dimension $m$. Let $ \{e_1, \cdots , e_m\} $ be a ...
1
vote
0
answers
44
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Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?
Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...
0
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0
answers
57
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Calculating the principal curvature of the geodesic sphere of radius $r$ in the space of dimension $n$ and of constant sectional curvature $c$
I'm reading "Riemannian Geometry" by Manfredo P. do Carmo and I'm trying to calculate the principal curvature of the geodesic sphere of radius $r$ in the space of dimension $n$ and of ...
1
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0
answers
49
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Size of conformal factor under uniformisation
Consider closed orientable surfaces whose metrics are hyperbolic (i.e., $K=-1$) except in a region which is a hemisphere of a unit sphere attached to the hyperbolic region along a closed geodesic (of ...
3
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0
answers
102
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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.
...
3
votes
0
answers
159
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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 $...
0
votes
1
answer
82
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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 ...
1
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0
answers
153
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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 ...
3
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0
answers
56
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Isometric embedding of 4-element metric spaces into Riemannian manifolds and the curvature
I came across this question Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension. In one of the answers it was stated that it is always possible to isometrically ...
0
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2
answers
206
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Gaussian curvature and curvature of the Levi-Civita connection
In a Riemannian surface $(S,g)$ consider the Levi-Civita connection $\nabla$ corresponding to the metric $g$. Suppose we have an orthonormal frame $\{e_1,e_2\}$ with dual coframe $\{\omega^1,\omega^2\}...
0
votes
1
answer
114
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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, ...
1
vote
0
answers
119
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Curvature operator on Kahler manifolds
Is positive curvature operator on a Kaehler manifold equivalent to the curvature operator being positive on real $(1, 1)$-forms? How do these conditions translate into the components of the curvature ...
1
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0
answers
58
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Existence of solution to prescribed curvature problem with given asymptotic on the punctured unit disc
I have trouble understanding a conclusion in the following paper: Prescribed Curvature and Singularities of Conformal Metrics on Riemann Surfaces by Robert C. McOwen
In the appendix, part B, we are ...
6
votes
2
answers
520
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Holonomy as integration of curvature for principal $G$-bundles?
Holonomy and curvature may seem to be slightly advanced topics in
geometry. However, their origins are easily imaginable. Namely,
picture the surface of earth $S$, and pick an arbitrary
contractible ...
1
vote
1
answer
122
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Requirement of parametrization of surfaces
If I have a smooth surface $M$ (2D embedded in 3D), under what conditions I can assure that there exists a finite collection of charts $\{U_i, \phi_i \}_{i}$, with $\phi_i : U_i \to M$, such that its ...
5
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0
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99
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What normal variational vector fields are allowed for the area to be preserved?
Let $(M^3,g)$ be a Riemannian manifold with boundary and let $\varphi : \Sigma \to M$ be a two-sided proper embedding of a compact surface with boundary, with unit normal $N$.
If $\varphi$ is not ...
18
votes
1
answer
1k
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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, ...
1
vote
0
answers
97
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Curvature of randomly generated B-spline curve
I am working on Bayesian statistical estimation of parameters (control points) of closed B-spline curve bounding an object on a an image. The problem is that I require those curves to not be much &...
5
votes
0
answers
187
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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
67
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Curvature estimate in terms of the boundary
The curvature of a minimal disc $S^2 \subset \mathbf{R}^3$ can be bounded in terms of the curvature of its boundary via the Gauss–Bonnet formula:
\begin{equation}
\frac{1}{2}\int_S \lvert A \rvert^2 \...
2
votes
0
answers
103
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 ...
1
vote
1
answer
113
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Effect of changing intersection normal curvatures on Gauss curvature $K$
The 30 straight edges of an icosahedron (with constant Euclidean vertex to vertex distance, and constant sphere center to vertex distance) have normal curvatures $\kappa_n=0$ in radial planes. They ...
1
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0
answers
96
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Does nefness in analytic setting depend on Hermitian metric?
I'm reading Demailly's book 'Analytic Methods in Algebraic Geometry'.
Let $X$ be a compact complex manifold with a Hermitian metric. A line bundle $L$ is said to be nef if for every $\epsilon>0$, ...
5
votes
2
answers
314
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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} =...
6
votes
1
answer
573
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Does every ‘curvature’ tensor induce a metric? [duplicate]
So we know that given a Riemannian manifold $(M,g)$ we can calculate its Riemannian curvature $R$. This covariant $4$-tensor field then satisfies some important symmetries
\begin{gather*}
R_{ijkl} = - ...
4
votes
1
answer
362
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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 ...
2
votes
0
answers
75
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Model structure for dga of (endormorphism) vector bundle valued differential forms
I was browsing and came across this discussion on the model structure for a dga. They mostly explain the commutative case but then say some things about the non-commutative case.
Context Consider a ...
9
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1
answer
336
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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 $K \to +\infty$...
1
vote
2
answers
217
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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 ...
1
vote
2
answers
143
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 ...
3
votes
0
answers
118
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Curvature explosion and metric landmark stability
$\newcommand{\prin}{\mathrm{prin}}$Context: Let $S$ be the unit sphere in some finite dimensional vector space $V$. Given a connected compact Lie group real representation $\rho:G\rightarrow O(V)$, ...
0
votes
0
answers
183
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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 ...
3
votes
1
answer
172
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 ...
6
votes
1
answer
371
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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-...
5
votes
1
answer
377
views
Compact complex non-Kähler manifolds with nef canonical bundle
Are there examples of compact complex manifolds $X$ with $K_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples?
Here, nef can be defined as follows: For any $\varepsilon>0$ there is ...
2
votes
0
answers
152
views
Expository material on the Gromov-Lawson surgery theorem
I am looking for an expository text on the paper "The classification of simply connected
manifolds of positive scalar curvature" by Gromov and Lawson, in particular on the proof of Theorem A....
2
votes
0
answers
170
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Asymptotic expansion of the Hessian of the distance function
This question originates from another question. Big thanks to MySheperd whose answer to that question clarified my thoughts.
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold and $r$ is ...
2
votes
0
answers
42
views
Scalar curvature of homogeneous bounded domains
Is it true that a bounded domain $\Omega \subset \mathbb C^n$ equipped with a homogeneous metric (not necessarily a Bergman metric) has (constant) negative scalar curvature?
4
votes
1
answer
1k
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Hessian of the distance function--comparison with the space form with constant sectional curvature 0
Let $M$ be an $n$-dimensional complete Riemannian manifold and $r$ is the distance function to a fixed point.
The Hessian comparison theorem says that if the sectional curvature of $M$ is bounded (...
3
votes
0
answers
59
views
Definition of sectional curvature of graph and relation to smooth sectional curvature
Let $(M,g)$ be a Riemannian manifold.
Let $\tau$ be a triangulation, i.e. a simplicial complex together with a fixed homeomorphism to $M$.
By forgetting about all cells except $0$-cells and $1$-cells ...
6
votes
2
answers
400
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Is it known whether a closed simply-connected manifold of non-negative curvature admits positive Ricci?
It is discussed in this question whether a simply-connected closed Riemannian manifold with non-negative Ricci curvature admits positive Ricci curvature, and the answer appears to be "no, there ...
2
votes
1
answer
213
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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 ...