# Questions tagged [curvature]

The curvature tag has no usage guidance.

175
questions

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votes

**1**answer

135 views

### Product formula for Laplace de-Rham operator

Let $M$ be a Riemannian manifold with Laplace de-Rham operator $\Delta = (d + \delta)^2$. If $g$ is a smooth $k$-form, and $f$ is a smooth function, is there a simple formula for $\Delta(fg)$ when $k &...

**0**

votes

**0**answers

105 views

### Curvature without any reference to embedding or coordinates

On a 2 sphere, which is topologically just a compact simply connected 2 dim space, one can edow geometric structure by requiring each two geodesics have equidistant intersections.
Is this enough to ...

**4**

votes

**1**answer

150 views

### Positive scalar curvature on the total space of a circle bundle

Let $(\Sigma_\gamma,g)$ be a closed and orientable Riemannian surface of genus $\gamma \geq 1$, $(M^3,\tilde{g})$ be a closed, connected and orientable Riemannian $3$-manifold, and $\pi : M \to \...

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votes

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60 views

### 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 ...

**6**

votes

**2**answers

604 views

### 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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votes

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66 views

### Realization of a $\mathfrak{g}$-valued $2$-form as a curvature form

Consider the Lie group $S^1$. Recall that the associated Lie algebra is $\mathbb{R}$.
Let $M$ be a manifold. Consider the second de-Rham cohomology group $H^2(M,\mathbb{R})$.
Let $\Omega\in H^2(M,\...

**8**

votes

**0**answers

55 views

### Jacobi fields on non-geodesic curves

The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for ...

**6**

votes

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146 views

### What is the definition of homotopy flat connections?

What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra

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57 views

### Parallel transport of vector along piecewise smooth loop on high-dimensional manifold

In this https://math.stackexchange.com/questions/2568300/gauss-bonnet-like-statement-connecting-parallel-transport-and-curvature question, it was discussed that the rotation of a vector that is ...

**1**

vote

**1**answer

220 views

### Katz's paper on $p$-curvature – help with proof understanding

I am studying N. Katz's paper "Nilpotent connections and the monodromy theorem: applications of a result of Turrittin" where I found a fairly good account on $p$-curvatures.
I don't understand the ...

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votes

**0**answers

159 views

### Non-commutative analogue of a certain fact in differential geometry

In the literature, is there a non-commutative analogue of the fact that every Riemannian manifold whose isometry group has sharp dimension must be a constant curvature manifold?

**6**

votes

**1**answer

119 views

### Integrating the Riemann curvature tensor over a singular 2-disc

There's a classic characterization of the Riemann curvature tensor. Say, take a Riemann metric on an open subset $U$ of $\mathbb R^n$. Given a point $p \in U$ and two vectors $v,w \in T_p U$ you ...

**7**

votes

**1**answer

161 views

### Positively curved manifold with collapsing unit balls

Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and
$$
\inf_{p \in M} \text{Vol}_gB(p,1)=0?
$$

**5**

votes

**1**answer

136 views

### Positively curved metric with uniformly positive scalar curvature

Can we find a complete noncompact Riemannian manifold $(M^n,g)$ with bounded geometry satisfying the following conditions?
the curvature operator $Rm>0$;
the scalar curvature $R \ge 1$.
Notice ...

**4**

votes

**0**answers

105 views

### Scalar curvature in terms of second fundamental form, reference request

I would like to cite a reference for the following formula for scalar curvature:
If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...

**0**

votes

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39 views

### Gauss curvature of a fibre as a submanifold in a Riemannian warped product

Consider the Riemannian warped product $M^{n+1}=I\times\mathbb{S}^n$ with metric
\begin{align}
g=dt\otimes dt+f(t)^2g_{\mathbb{S}^n}
\end{align}
where $I\subseteq\mathbb{R}$ is some open interval and ...

**2**

votes

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63 views

### For a manifold of positive curvature, can we lower bound the distance between unit normals?

Suppose $M \subset \mathbb R^d$ is a $C^2$ $(d-1)$-manifold. In particular I am interested in when $M$ is the set $\{x \in \mathbb R^d: f(x) \le 1\}$ for some $C^2$ function $f:\mathbb R^d \to \...

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votes

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105 views

### 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 $L$ ...

**2**

votes

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89 views

### 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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vote

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64 views

### 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 ...

**3**

votes

**1**answer

162 views

### 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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votes

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92 views

### 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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vote

**1**answer

285 views

### 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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votes

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105 views

### 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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votes

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135 views

### Is there any geometric interpretation for the trace of Fisher information matrix?

Consider a parametric family $p_\theta(x)$ of distributions, with parameter $\theta \in \Theta \subseteq \mathbb R^p$.
If the mapping $\theta \mapsto p_\theta(x)$ is continuously differentiable at $\...

**3**

votes

**1**answer

135 views

### Upper bound on the sectional curvature of the orthogonal group

Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T_A O(n) =...

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votes

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202 views

### Conventions for Riemann curvature tensor

I am aware of two conventions for the Riemann curvature tensor, namely the expression
$$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$
is either declared to be $R(X,Y,Z,W)$ or $...

**7**

votes

**2**answers

220 views

### Is every metric uniformly close to a metric with negative scalar curvature?

Let $M$ be a smooth manifold with non-empty boundary.
Let $g$ be a smooth Riemannian metric on $M$. Is the following true?
For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ ...

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votes

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77 views

### Positive scalar curvature and $\mathbf{H}P^2$-bundles

Let $M$ be a simply-connected spin-manifold of dimension $n\geq 5$. The Atiyah-Bott-Shapiro orientation $\mathrm{MSpin} \to KO$ produces an element $\alpha(M)$ of $\pi_n KO$. Results of Gromov-Lawson ...

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vote

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83 views

### Natural measures of curvature of Riemannian manifold along two-dimensional subspace

Given a Riemannian manifold $M$, a point $p \in M$, and some two-dimensional subspace $\varSigma$ of $T_{p}M$, the sectional curvature $K(\varSigma)$ is a well-known, natural measure of the curvature ...

**5**

votes

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128 views

### Effect of the inverse exponential map on the curvature of a given curve

Suppose you have a curve $\alpha$ in a manifold $\mathcal{M}$. You are at a point $\alpha(t)$ of that curve. The curvature of $\alpha(t)$ is the same as the curvature of the curve $exp^{-1}_{\alpha(t)}...

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votes

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106 views

### Flatness equivalence

Let $\pi:E\rightarrow M$ be a complex vector bundle and $H$ a hermitian metric over it. If $D$ is a connection over $E$, using the metric $H$, we can decompose it as:
$$
D=D_H+\phi
$$
Where $D_H$ is ...

**3**

votes

**0**answers

70 views

### 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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vote

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141 views

### 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 ...

**9**

votes

**0**answers

183 views

### Hermitian sectional curvature

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.
...

**5**

votes

**1**answer

241 views

### Injectivity radius on complete manifolds with positive and bounded curvature

I have two question:
1) Are there any examples of complete manifold with strictly positive and bounded section curvature which has zero injectivity radius?
2) Is there a sequence of non-compact ...

**1**

vote

**1**answer

177 views

### When are principal lines of curvature geodesics?

Let $S$ be a smooth surface embedded in $\mathbb{R}^3$.
When are (some of) the principal lines of curvature geodesics
on $S$? Perhaps on the ellipsoid below, the (blue) central
cycle, a max principal ...

**1**

vote

**0**answers

69 views

### Is the Frenet frame is independent of the choices of parameters?

I asked this question on StackExchange, but until now there is not any answer or hint. I hope I can get some help here.
When I am reading ''A course in differential geometry'' of Klingenberg, I ...

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votes

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170 views

### Formula for difference between curvature operators?

This is a re-editing of a prerviously posted question:
Let $(M,g)$ be a Riemannian manifold. Let $C:TM\to TM$ be symmetric positive definite. Define the metric
$$
(X,Y)_C = (X,CY)_g.
$$
Denote by $\...

**6**

votes

**1**answer

232 views

### 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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vote

**1**answer

154 views

### determinant of curvature (notation issue)

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 ...

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votes

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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 ...

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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 ...

**3**

votes

**1**answer

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 ...

**4**

votes

**1**answer

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?

**5**

votes

**1**answer

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 (...

**4**

votes

**1**answer

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 ...

**3**

votes

**1**answer

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 $...

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votes

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85 views

### Positive and non-negative sectional curvature of semi-riemannian metrics

I am currently studying the techniques related to geometry of positive and non-negative sectional curvature of Riemannian metrics. In particular, I have done some work with Cheeger deformations. I ...

**3**

votes

**1**answer

110 views

### Flatness in a neighborhood of a point condition

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 ...