All Questions
Tagged with riemannian-geometry differential-topology
124 questions
3
votes
2
answers
247
views
Morse approximation with bounded number of critical points
Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
- 716
3
votes
0
answers
49
views
Transport map to lower dimension?
Let $S^{d-1}$ be the sphere in $\mathbb{R}^d$.
Given a $C^\infty$ function $f \colon S^{d-1} \to \mathbb{R}$, define $g \colon S^{d-1} \to S^{d-1}$ as $g(x) = \exp_x(\nabla f(x))$, where $\nabla f(x)$ ...
- 171
28
votes
3
answers
2k
views
Does isometric immersion map boundary to boundary?
Let $M$ be a compact, connected, oriented, smooth Riemannian manifold with non-empty boundary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion.
Is it true that $f(\partial M) \...
- 356
2
votes
0
answers
82
views
Is isoperimetric hypersurface unique up to homeomorphism?
Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\...
- 356
4
votes
1
answer
179
views
Convex hull and least area discs in Riemannian 3-manifolds
$\DeclareMathOperator\Conv{Conv}$Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all ...
CommunityBot
- 1
0
votes
0
answers
27
views
Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$
It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
- 383
0
votes
1
answer
91
views
On nontrapping manifolds
Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary.
Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is ...
- 39k
7
votes
2
answers
614
views
Locally conformally flat
Is there any example of a locally conformally flat manifold that is neither a space form nor a product of space forms?
- 29.1k
2
votes
1
answer
122
views
Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator
Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator.
My question: Is there always a smooth spin map ...
- 19.3k
3
votes
2
answers
604
views
Calculation of the top Chern class of spinor bundle over $S^{2n}$
It's well known that for a complex vector bundle $E$, we have
$$c_n(E)=e_n(E_\mathbb{R}) $$
But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...
- 9,853
16
votes
0
answers
425
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 587
0
votes
0
answers
183
views
Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$.
On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
- 144
2
votes
0
answers
211
views
When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
15
votes
6
answers
2k
views
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those ...
4
votes
1
answer
202
views
Finding a real-analytic diffeomorphism
Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
- 2,479
6
votes
1
answer
230
views
Vanishing directional derivatives on $S^2$
Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(...
- 7,197
2
votes
0
answers
127
views
Foliation of $X$ by once punctured planes without any singularities
Let $n=3.$
Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-...
- 383
18
votes
1
answer
1k
views
Is the minimal volume a topological invariant?
On Wikipedia, it is said that the minimal volume
$$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$$
is a topological invariant, ...
- 31.5k
4
votes
1
answer
216
views
In what topology does Gromov's lemma hold on noncompact symplectic manifolds?
In symplectic geometry, it is commonly said that ``the set of almost complex
structures tamed to a symplectic form is contractible'' even on noncompact symplectic manifolds. In my understanding
one ...
- 17.5k
4
votes
1
answer
258
views
Is every Riemannian metric conformally equivalent to one that is geodesically complete?
The question in the title seems a natural one to ask, and I suspect that it is already considered, even completely solved, somewhere in the literature. Although I prefer some explanation of the idea(s)...
- 829
4
votes
1
answer
290
views
Can every diffeomorphism be rescaled into a volume preserving one?
This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism.
Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ...
- 45k
4
votes
1
answer
334
views
Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief ...
- 4,713
5
votes
2
answers
769
views
Checking that the image of a curve is not contained in a hyperplane
Let $\gamma : [0,1] \to \mathbb R^n$ be a smooth curve, $n \geq 2$. I would like to find an easy to check condition such that the image of $\gamma$ is not contained in an $n-1$ dimensional linear ...
- 45k
3
votes
0
answers
608
views
Show that continuous maps between smooth manifolds can be approximated by smooth maps WITHOUT using Whitney's embedding theorem
As it is well-known (and for example this question shows) each continuous map between smooth manifolds is homotopically equivalent to a smooth map that can be constructed using the Whitney embedding ...
- 1,149
1
vote
0
answers
166
views
Is the standard $\mathbb R^4$ the only one with positive sectional curvature?
Perelman--Cheeger--Gromoll Soul Theorem states that whenever a complete non-compact Riemannian manifold $(M,g)$ has positive sectional curvature, it should be diffeomorphic to an Euclidean Space.
On ...
- 1,774
1
vote
2
answers
148
views
Construct a hypersurface with fixed principal curvatures at a point
I'm reading Eschenburg's paper Local convexity and nonnegative curvature —
Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M$, $N\in T_pM$, we want to ...
- 27.5k
4
votes
1
answer
131
views
Smooth circle actions on Riemannian manifolds and harmonicity of quotient map
Let $(M, g)$ be a compact Riemannian manifold. Suppose $M$ admits a smooth free circle action (Denote the circle group by $G$. The action $G$ on $M$ is not necessarily isometric) and the orbit space $...
- 7,583
2
votes
2
answers
387
views
Under what conditions can an orientable Riemannian 3-manifold be defined implicitly?
Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly?
What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such ...
- 68.9k
6
votes
4
answers
3k
views
Does every smooth manifold of infinite topological type admit a complete Riemannian metric?
To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric ...
- 2,821
8
votes
0
answers
409
views
What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?
In his paper [2], Paul Ehrlich write
In [1], Aubin stated a theorem which implied as a corollary that if a manifold
$M$ admits a Riemannian metric with nonnegative Ricci curvature and
all Ricci ...
- 4,195
2
votes
1
answer
119
views
Density of smooth bi-Lipschitz maps in smooth maps
Setup/Motivation:
Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class ...
- 26.3k
6
votes
2
answers
435
views
The convex hull of a manifold whose cobordism class is trivial
Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...
- 6,367
2
votes
0
answers
137
views
Question about spin map
I'm confused with the following definition of a spin map.
A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...
- 563
1
vote
0
answers
116
views
Existence of a local spinor bundle
I am confused about the existence of a local spinor bundle.
My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...
- 563
2
votes
1
answer
256
views
Equidistant points on a compact Riemannian manifold
Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows:
$K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at ...
- 10.6k
-1
votes
1
answer
859
views
Essential simple closed curves in a torus [closed]
Definition: By a closed curve in a surface $S$ we will mean a continuous map $S^1 \to S$.
We will usually identify a closed curve with its image in $S$. A closed curve
is called essential if it is not ...
- 28.2k
7
votes
1
answer
367
views
Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^...
- 1,424
6
votes
0
answers
341
views
When exponential map is 1-1 from vector fields to diffeomorphisms
Let $M$ be a connected and complete Riemannian manifold of positive dimension, $k$ be a positive integer, and let $\mathfrak{X}^k_c$ be the set of class $C^k$-vector fields on $M$ of compact support. ...
- 10.4k
2
votes
0
answers
70
views
Compatible almost complex structures such that the associated riemannian metric has positive injectivity radius
Let $M$ be a compact manifold, consider $\omega$ the canonical symplectic form in $T^*M$ and $\hat J$ the canonical almost complex structure coming from the Sasaki metric.
Let $\mathcal{J}$ be the set ...
- 791
1
vote
0
answers
93
views
A question about Homotopy equivalence (II)
I posted a similar but different question before in the link
https://math.stackexchange.com/questions/4311982/why-does-x-0-times-s1-simeq-x-x-0/4312530?noredirect=1#comment8987557_4312530.
Now, my new ...
- 563
0
votes
1
answer
154
views
Why does $X_0\times S^1\simeq X-X_0$? [closed]
Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion?
$$X_0\...
- 28.2k
6
votes
0
answers
302
views
Are there are any surprising diffeomorphisms?
Two smooth manifolds are often viewed to be equivalent if there is a diffeomorphism between them. Are there examples of two manifolds that one would not expect to be equivalent (in this sense), but in ...
- 1,379
3
votes
6
answers
2k
views
The purpose of connections in differential geometry [closed]
I am currently reading through differential geometry as a mathematics graduate.
Can somebody give me a brief explainer on the purpose of connections?
I could also use explainers on differential forms. ...
- 26.3k
2
votes
2
answers
523
views
Orthogonal smooth vector field on a Riemannian manifold
Consider a compact Riemannian manifold $M$ with a smooth metric, and a smooth vector field $X$ on $M$. My question is, when can we construct another smooth vector field $Y$ on $M$ such that $Y$ is ...
- 44.7k
16
votes
1
answer
1k
views
Is there an area-preserving diffeomorphism of the disk which is nowhere conformal?
This question is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk.
Does there exist a smooth area-preserving diffeomorphism $f:D \to D$ that does not have conformal points?
I ...
- 6,741
10
votes
2
answers
926
views
Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?
This is a cross-post. While working on a variational problem, I have reached to the following question.
Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...
- 134
5
votes
1
answer
597
views
A vector field whose flow has constant singular values
$\newcommand{\tr}{\operatorname{tr}}$
$\renewcommand{\div}{\operatorname{div}}$
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi_t$ be its flow.
Does ...
- 108k
4
votes
1
answer
211
views
Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology
Cross-post from MSE.
Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. ...
- 108k
10
votes
2
answers
1k
views
Metric deformations from non-negative to positive curvature
Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g_1$ satisfying $\mathrm{sec}(M,g_1) ...
- 63.9k
6
votes
1
answer
229
views
Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Cross post from MSE. and sorry if this is an obvious question.
Here is a line of proof of Theorem 1.15 from
Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. ...
- 4,195