Questions tagged [curvature]
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
282
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Injectivity radius bound for a metric with bounded curvature on $\mathbb{R}^n$
My question is as follows:
Question: Is it true that if $g$ is a metric (need not be complete) on $\mathbb{R}^n$ such that $B_g(x_0, 1)\subset \subset \mathbb{R}^n$, and $g$ has bounded curvature on a ...
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1
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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 ...
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7
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What is the best way to draw curvature?
This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures ...
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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 ...
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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 ...
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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 ...
- 51
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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 ...
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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 ...
- 3,192
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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 ...
- 3,493
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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 ...
- 153
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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 ...
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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 ...
- 141
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1
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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, ...
CommunityBot
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3
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Geometric picture of scalar curvature
In first course differential geometry you learn, that Ricci-curvature is something like a mean-value of the curvature endomorphism, because it's a trace, and the scalar curvature is again a mean-value ...
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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 ...
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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 ...
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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 ...
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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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Small perimeter-minimizing disks on curved surfaces
Suppose that I have a smooth curved surface, and I choose an arbitrary point $Q$ on that surface. Say the Gaussian curvature at that point is $K$. What I am wondering is, is there an expression for ...
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How many minimal surfaces do we have if the metric in the target space is not flat
It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point,
and any two othogonal vectors in this plane, and any ...
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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.
...
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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 $...
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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 ...
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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 $K \to +\infty$...
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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 ...
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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\}...
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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 ...
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Rigidity for convex surfaces in elliptic/hyperbolic space
From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
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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 ...
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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 ...
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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 ...
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1
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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 ...
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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 ...
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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, ...
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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 &...
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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 ...
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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 \...
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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 ...
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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 $\epsilon>0$, ...
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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} =...
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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} = - ...
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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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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 ...
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On the crookedness of curves (Milnor's paper)
I am reading the paper (Ann Math. 1950) "on the total curvature of knots" by J. Milnor.
I was trying to understand this part of the paper (here is a free access link to the paper):
Let $P$ be a ...
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On the the number of intersections of a knotted polygon with a plane.(Milnor's paper)
I'm trying to understand the article "on the total curvature of knots" by John. W. Milnor. here is the free access to the article .
the last theorem in this paper indicates that for every knotted ...
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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 ...
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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 ...
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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 ...
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2
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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 ...
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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 $\rho:G\rightarrow O(V)$, ...
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