# Questions tagged [curvature]

Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)

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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 ...
93 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. ...
150 views

83 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, ...
1 vote
104 views

### 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 vote
53 views

### 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 ...
349 views

### 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
104 views

### 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 ...
97 views

### 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 ...
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, ...
1 vote
58 views

### 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 &...
159 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 ...
66 views

### 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 \...
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 ...
1 vote
75 views

### 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 ...
1 vote
88 views

### 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 $\epsilon>0$, ...
296 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} =...
445 views

### 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} = - ...
291 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 ...
68 views

### 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 vote