The curvature tag has no usage guidance.

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### holomorphic sectional curvature and total scalar curvature

In a paper of Heier and Wong, It is written that from a pointwise argument due to Berger does follow that the scalar curvature (and thus also the total scalar curvature) of a Kaehler metric of ...

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votes

**1**answer

164 views

### volume of a submanifold implies bounds on curvature

I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred ...

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**0**answers

257 views

### Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...

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vote

**3**answers

325 views

### Surface analog of clothoid: curvatures covering $\mathbb{R}$

The clothoid $C$, a.k.a. the Euler spiral,
is one among many curves with
the property that its curvatures cover $\mathbb{R}$
in the sense that, for every $x \in \mathbb{R}$,
there is a point $p \in C$ ...

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**0**answers

235 views

### Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...

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votes

**2**answers

326 views

### Can one (block) diagonalize the curvature matrix of 2 forms on a Riemannian manifold?

Let $M$ be a smooth Riemannian manifold, let $R$ be the Riemannian curvature operator, and let $p$ be a point in the manifold. With respect to any orthonormal basis of the tangent bundle at the point $...

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votes

**1**answer

686 views

### Curvature of contour lines of a scalar field

How can I compute the curvature of the contour lines (equipotential lines) $\phi (\vec{r})=c$ for the scalar field $\phi (\vec{r})$ ? I expect the direction of the curvature vector to be along the ...

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votes

**1**answer

720 views

### About Sectional Curvature [closed]

In a paper by Yann Ollivier:
Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint
of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...

**3**

votes

**1**answer

349 views

### geometric meaning of Ricci-flatness

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...

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votes

**3**answers

2k views

### 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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votes

**2**answers

245 views

### Is there a combinatorial analogue of the Kazdan Warner theorem?

First let me state a result of Kazdan and Warner
Let $M$ be a compact orientable two dimensional manifold.
Let $f:M \rightarrow \mathbb{R}$ be a function that has the same
sign as the Euler ...

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votes

**2**answers

1k views

### A question on Ricci curvature and Ricci form.

It seems to me that there are two definition of Ricci flatness; real and complex ones. The real Ricci flatness claims vanishing of Ricci curvature. To define complex Ricci flatness, one needs to ...

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votes

**1**answer

432 views

### Surfaces in a 3-manifold with the same Gaussian curvature with respect to two ambient conformal metrics

Let $M$ be a 3-smooth manifold and $g_{1}$ and $g_{2}$ two conformal metrics on $M$. Consider an immersed surface S in $M$ and let $K_{1}$ and $K_{2}$ be the Gaussian curvatures of $S$ with respect to ...

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**0**answers

152 views

### Strake's splitting theorem for complex sectional curvature

Strake's splitting theorem: Let $(M^n,g)$ be an open manifold with sectional curvature $K\ge 0$ and soul $\Sigma^k$. If the normal holonomy group of $\Sigma$ is trivial, $M$ splits isometrically into $...

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votes

**1**answer

198 views

### curvature of curves in the space of gaussians measures

I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider this sequence to be ...

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votes

**3**answers

793 views

### Open problems in sub-Riemannian geometry

What are some open problems in sub-Riemannian geometry?
I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.

**6**

votes

**1**answer

372 views

### When is the hull of a space curve composed of developable patches?

Let $C$ be a smooth curve in $\mathbb{R}^3$ that lies entirely on its convex hull,
$\cal{H}(C)$.
Under what conditions on $C$ is $\cal{H}(C)$ the union of developable surface patches?
I believe ...

**3**

votes

**2**answers

1k views

### Interpretation of Curvature and Torsion

Dear all,
When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields
$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\...

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votes

**1**answer

689 views

### Is there a mathematical explanation for the Aharonov-Casher effect?

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows.
Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...

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votes

**3**answers

1k views

### History surrounding Gauss Theorema Egregium and differential geometry

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...

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votes

**1**answer

181 views

### Flattening a corner in a convex $d$-polytope (into $d-1$ dimensions, without overlap)?

I'm interested in the following question, which seems to be assumed all over the place (at least for 3 dimensions) in convex geometry, and which I cannot find a proof of.
Suppose we have a corner ...

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votes

**0**answers

189 views

### Manifolds with a lower degree of regularity

I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below).
There, the authors consider domains of $R^n$ with regularity of class $W^2 L^{n-1,1}$(...

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votes

**3**answers

3k views

### Curvature of a Lie group

Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...

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votes

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3k views

### When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...