Questions tagged [curvature]

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

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Algebraic content of Gauss's Theorema Egregium

Let me first recall what Gauss's Theorema Egregium says. Consider a surface isometrically embedded in $\mathbb{R}^3$. In some local coordinates, let the first and second fundamental forms be $$E \...
Tara's user avatar
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5 votes
1 answer
254 views

Manifolds with boundary admitting no closed embedded minimal hypersurface

The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex ...
Eduardo Longa's user avatar
17 votes
1 answer
888 views

Geometric interpretation of the Weyl tensor?

The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops. Question: Is there a similarly direct geometric ...
Tim Campion's user avatar
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1 vote
1 answer
207 views

Wasserstein space with strictly non-positive sectional curvature

Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$. Does it ...
pseudocydonia's user avatar
3 votes
1 answer
350 views

Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds

In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully": Suppose $(X,g_X)$ and $(Y,g_Y)$ are ...
Lawrence Mouillé's user avatar
0 votes
0 answers
91 views

mean curvature for codimension $>1$?

The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
Johnny T.'s user avatar
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3 votes
1 answer
433 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 &...
Nathanael Schilling's user avatar
4 votes
1 answer
275 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 \...
Eduardo Longa's user avatar
4 votes
0 answers
97 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 ...
Ivan Meir's user avatar
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6 votes
2 answers
741 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. ...
Plank's user avatar
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2 votes
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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,\...
Praphulla Koushik's user avatar
12 votes
0 answers
247 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 ...
Ethan Dlugie's user avatar
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6 votes
1 answer
335 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
Jim Stasheff's user avatar
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4 votes
0 answers
180 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 ...
Foivos's user avatar
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2 votes
1 answer
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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 ...
Conjecture's user avatar
4 votes
0 answers
241 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?
Ali Taghavi's user avatar
6 votes
1 answer
311 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 ...
Ryan Budney's user avatar
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7 votes
1 answer
193 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 $$ \inf_{p \in M} \text{Vol}_gB(p,1)=0? $$
Totoro's user avatar
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5 votes
1 answer
156 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 ...
Totoro's user avatar
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4 votes
0 answers
634 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 ...
Yasha Berchenko-Kogan's user avatar
0 votes
0 answers
54 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 ...
Anonymous amateur's user avatar
2 votes
0 answers
89 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 \...
Daron's user avatar
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2 votes
0 answers
149 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 $L$ ...
user267839's user avatar
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6 votes
1 answer
322 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 ...
L.F. Cavenaghi's user avatar
2 votes
0 answers
89 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 ...
BVquantization's user avatar
4 votes
1 answer
442 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 ...
Math_tourist's user avatar
6 votes
0 answers
110 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 ...
Tyrone's user avatar
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3 votes
1 answer
1k 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 ...
Overflowian's user avatar
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4 votes
0 answers
126 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 ...
Nobody's user avatar
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2 votes
0 answers
633 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 $\...
dohmatob's user avatar
  • 6,716
3 votes
1 answer
578 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) =...
Călin's user avatar
  • 271
6 votes
0 answers
867 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 $...
John Pardon's user avatar
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7 votes
2 answers
384 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 $g_{\epsilon}$ ...
Asaf Shachar's user avatar
  • 6,611
2 votes
0 answers
97 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 ...
skd's user avatar
  • 5,550
1 vote
0 answers
92 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 ...
Matteo Raffaelli's user avatar
6 votes
0 answers
265 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)}...
Damaru's user avatar
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4 votes
0 answers
118 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 ...
Leonardo Schultz's user avatar
3 votes
1 answer
122 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 ...
Amirhossein's user avatar
1 vote
1 answer
224 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 ...
Amirhossein's user avatar
9 votes
0 answers
274 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. ...
seub's user avatar
  • 1,337
8 votes
1 answer
969 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 ...
Yuchen Bi's user avatar
  • 101
1 vote
1 answer
565 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 ...
Joseph O'Rourke's user avatar
1 vote
0 answers
148 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 ...
Nan's user avatar
  • 11
6 votes
0 answers
196 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 $\...
Raz Kupferman's user avatar
6 votes
1 answer
474 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 ...
Jae Ho Cho's user avatar
1 vote
1 answer
204 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 ...
Praphulla Koushik's user avatar
5 votes
0 answers
225 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 ...
Leonardo's user avatar
  • 395
2 votes
0 answers
205 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 ...
Ali Taghavi's user avatar
3 votes
1 answer
106 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 ...
Andy Mack's user avatar
  • 255
4 votes
1 answer
247 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?
Minghui Ouyang's user avatar