Questions tagged [curvature]

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3
votes
1answer
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
0answers
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
1answer
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 \...
4
votes
0answers
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
2answers
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. ...
2
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0answers
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
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0answers
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
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0answers
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
3
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0answers
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
1answer
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 ...
2
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0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
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0answers
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
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0answers
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 \...
2
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0answers
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
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0answers
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 ...
1
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0answers
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
1answer
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 ...
5
votes
0answers
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 ...
1
vote
1answer
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 ...
4
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0answers
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 ...
2
votes
0answers
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
1answer
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) =...
3
votes
0answers
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
2answers
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 $g_{\epsilon}$ ...
2
votes
0answers
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 ...
1
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0answers
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
0answers
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)}...
4
votes
0answers
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
0answers
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 ...
1
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0answers
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
0answers
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
1answer
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
1answer
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
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0answers
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 ...
6
votes
0answers
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
1answer
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 ...
1
vote
1answer
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 ...
5
votes
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 ...
2
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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 ...
3
votes
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 ...
4
votes
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?
5
votes
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 (...
4
votes
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 ...
3
votes
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 $...
4
votes
0answers
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
1answer
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 ...