Questions tagged [curvature]
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
273
questions
1
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0
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45
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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 ...
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3
votes
0
answers
93
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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.
...
- 295
3
votes
0
answers
150
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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 $...
- 895
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votes
1
answer
73
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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 ...
- 85
1
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0
answers
105
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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 ...
- 1,217
3
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0
answers
51
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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 ...
- 748
0
votes
2
answers
164
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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
83
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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, ...
- 111
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0
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104
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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 ...
- 71
1
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0
answers
53
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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
349
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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 ...
- 4,648
1
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1
answer
104
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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 ...
- 13
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0
answers
97
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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 ...
- 2,023
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, ...
- 559
1
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0
answers
58
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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 &...
- 21
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159
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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
66
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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 \...
- 4,636
2
votes
0
answers
79
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 ...
- 3,883
1
vote
1
answer
75
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Effect of changing intersection normal curvatures on Gauss curvature $K$
The 30 straight edges an icosahedron ( constant Euclidean vertex to vertex distance, constant sphere center to vertex distance ) have normal curvatures $ kn=0 $ in radial planes). They span and ...
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88
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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 $\epsilon>0$, ...
- 283
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2
answers
296
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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} =...
- 103
5
votes
1
answer
445
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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} = - ...
- 103
4
votes
1
answer
291
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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 ...
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0
answers
68
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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 ...
- 1,331
1
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0
answers
232
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Second fundamental form trace and constrained curvature
The mean curvature for submanifolds with higher codimension (i.e., $\ge 1$), it is defined as the trace of the second fundamental form scaled by the inverse of the dimension of the submanifold, then:
$...
- 305
9
votes
1
answer
324
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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 $K \to +\infty$...
- 4,636
1
vote
2
answers
183
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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 ...
- 423
1
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2
answers
141
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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
3
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0
answers
117
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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 $\rho:G\rightarrow O(V)$, ...
- 59
0
votes
0
answers
114
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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
169
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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 ...
- 213
6
votes
1
answer
309
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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-...
- 8,965
5
votes
1
answer
344
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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 ...
- 255
2
votes
0
answers
117
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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....
- 147
2
votes
0
answers
100
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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 ...
- 1,309
2
votes
0
answers
39
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?
- 81
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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 (...
- 1,309
3
votes
0
answers
56
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 ...
- 557
6
votes
2
answers
347
views
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 ...
- 123
2
votes
1
answer
210
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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 ...
- 713
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0
answers
98
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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 ...
- 2,023
2
votes
0
answers
98
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The Ricci curvature is bounded below by scalar curvature
So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...
- 263
5
votes
1
answer
210
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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 ...
- 305
1
vote
1
answer
126
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Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous
This is followup to the following question: On the Lipschitz continuity of the unit-normal vector field of a polytope
Let $C$ be a (nonempty) closed convex subset of $\mathbb R^n$. Note that to every ...
- 6,466
1
vote
1
answer
146
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Conditions for Lipschitzness of boundary normal vector, almost everywhere
Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition
(Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique ...
- 6,466
5
votes
1
answer
228
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 ...
- 305
0
votes
0
answers
100
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Knots with everywhere positive curvature
A naive question that my searches have not resolved:
Q. Can every knot be realized by a curve in $\mathbb{R}^3$ with strictly positive
curvature at every point?
- 148k
3
votes
0
answers
92
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 ...
1
vote
1
answer
201
views
Heat kernel on hyperbolic space of variable curvature
I am working with the heat kernel on the hyperbolic space explicitly (as you may guess by my previous questions) and I got the desired results when the curvature is $-\kappa=-1$. Now I am trying to do ...
- 163
16
votes
1
answer
1k
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Gauss-Bonnet Theorem: Neither Gauss nor Bonnet [closed]
Tristan Needham says (p.174),*
"While Gauss and Bonnet certainly paved the road to [the Gauss-Bonnet Theorem],
neither one of them was even aware of this extraordinary result, let alone stated ...
- 148k