# Questions tagged [curvature]

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### Closed curves with minimal total curvature in the unit circle

Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L. In this later paper Chakerian gave a simpler proof and noted that ...
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### 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. ...
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### Construct Torsion element in $H^2(X,\mathbb{Z})$ with Ambrose–Singer theorem

Let $X$ be a Kahler manifold. Using the exponential sequence one obtains a homomorphism $c_1:H^1(X,\mathcal{O}_X^*)\rightarrow H^2(X,\mathbb{Z})$. This is associating to a holomorphic line bundle $L$ ...
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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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### Poincaré connection encode torsion and curvature

I'm trying to understand something that is written in Baez & Wise paper "Teleparallel Gravity as a Higher Gauge Theory". In section 4, they discuss Poincaré connection and, first of all, split ...
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### Curvature estimate for minimal surfaces

I am a bit confused about Theorem 2.16 in the book "A Course in Minimal Surfaces" by Colding and Minicozzi. The authors write that Theorem 2.16 was proved in this paper by Schoen and Simon in the more ...
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### 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 ...
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### Flat connections, curvature and holonomy

Let $A$ be a flat connection on a principal $G$-bundle $G\hookrightarrow P\to M$. Consider an homotopically trivial loop $\gamma \subset M$. For simplicity, suppose $\gamma = \partial D$ is the ...
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### Area lower bound given a mean curvature upper bound?

If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...
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### Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature

Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
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This is when studying about Chern classes from Kobayashi and Nomizu. Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$. Let $p:P\rightarrow ... 0answers 159 views ### Explicit parametrization of closed space curves of constant curvature Joining arcs of helices it is pretty easy to obtain closed curves with constant curvature and$C^2$regularity (see http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0203.pdf). Joining arcs of Salkowski ... 0answers 154 views ### A geometric rank of Riemannian manifolds There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks: The maximum number of global independent vector fields which can be defined ... 1answer 97 views ### Sectional curvatures under simple maps Suppose that we have a submanifold$X$of$\mathbb{R}^n$with the induced Euclidean metric, whose sectional curvatures we have a handle of (say, they are lower bounded by some$\kappa$). Is there a ... 1answer 149 views ### A variation on four-vertex theorem Is it true that, if a closed, strictly convex curve has exactly four vertices (extrema of curvature), then any circle has at most four points of intersection with it? 1answer 311 views ### Principal curvatures of$\mathbb{R}^{n^2}$-embedded SO(n) It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula$k(X,Y) = \frac{1}{4} \|[X Y]\|^2$. I'm wondering if the following (... 1answer 215 views ### Realizing the cross product of$\mathbb{R}^3$as the curvature tensor of a Riemannian metric on$\mathbb{R}^3$Is there a Riemannian metric on$\mathbb{R}^3$for which the corresponding curvature tensor$R$satisfies$R(X,Y)Z=(X\wedge Y)\wedge Z$? I have already discussed this question in the following post ... 1answer 138 views ### Special spheres: principal curvatures with different signs For$\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface$M^n$of a riemannian manifold$(N^{n+1},g)$is$\varepsilon$-convex whenever all principal curvatures of$...
Suppose that we have a Riemannian Manifold $(M,g)$ whose curvature vanishes in an open neighborhood U of a point p. When does this imply that the metric is Flat ? In particular, does it happen ...