Questions tagged [curvature]
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
283
questions
13
votes
0
answers
667
views
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 \...
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 ...
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 ...
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 ...
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 ...
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 ...
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 &...
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 \...
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 ...
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. ...
2
votes
0
answers
108
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,\...
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 ...
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
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 ...
2
votes
1
answer
449
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 ...
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?
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 ...
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?
$$
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 ...
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 ...
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 ...
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 \...
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$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\...
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) =...
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 $...
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 $g_{\epsilon}$ ...
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 ...
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 ...
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)}...
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 ...
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 ...
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 ...
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.
...
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 ...
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 ...
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 ...
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 $\...
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 ...
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 ...
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 ...
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 ...
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 ...
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?