All Questions
Tagged with reference-request dg.differential-geometry
800 questions
3
votes
0
answers
188
views
References and results for the eigenvalues of Ricci tensor
I am looking for references or results that gives estimates for every eigenvalue of the Ricci tensor. For example, the least eigenvalue is related to the minimum of the Ricci curvature, what can we ...
- 1,774
4
votes
1
answer
301
views
Injectivity radius of parallel hypersurfaces
Let $(M,g)$ be a Riemannian manifold and let $N$ be a compact hypersurface isometrically embedded into $M$ and let $\eta$ denote a choice of unit normal vector field on $N$. It is then true that $N$ ...
- 409
4
votes
1
answer
486
views
Every _______ $d$-manifold has an $S$-structure
I am looking for some analogous nontrivial but known statements and references about statements of the form:
Every _______ $d$-manifold has an $S$-structure.
Here _______ is a placeholder for ...
- 10.5k
13
votes
1
answer
1k
views
Aleksandrov's proof of the second order differentiability of convex functions
Aleksandrov [A], proved a remarkable property of convex functions.
Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
- 28k
5
votes
0
answers
243
views
Reference request : Quotient manifold theorem for Lie groupoid action on a manifold
Let $G$ be a Lie group and $M$ be a smooth manifold. Let $G\times M\rightarrow M$ be a smooth map giving a free, proper action of $G$ On $M$. Then, by quotient manifold theorem, we see that there ...
- 6,023
4
votes
0
answers
880
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 ...
14
votes
4
answers
2k
views
Book on manifolds from a sheaf-theoretic/locally ringed space PoV
I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds ...
6
votes
3
answers
1k
views
The isometry group of a product of two Riemannian manifolds
Under what conditions is the isometry group of a product of two Riemannian manifolds the product of the isometry groups of each one of the components?
One counterexample is a product of two isometric ...
- 2,535
5
votes
1
answer
248
views
Multisignature and homeomorphism type
In classical surgery theory, there is a map
$$L_{n+1}(\pi_1M)\to S(M^n)$$
Element in $L_{n+1}(\pi_1M)$ is realized as surgery obstruction of a surgery problem to $M\times I$ with one boundary piece ...
- 101
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
- 622
3
votes
1
answer
270
views
survey paper on the construction of hyperbolic manifolds
Is there a good survey paper which discusses the common ways of building hyperbolic $n$-manifolds?
- 101
5
votes
0
answers
94
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Progress of the Kazdan-Warner Problem on Higher-genus Surfaces
I would like to understand if there is any further progress of the problem of prescribing Gaussian curvature on (oriented) closed surface $M$ with $\chi(M)<0$ in a conformal class after Kazdan and ...
- 402
6
votes
2
answers
755
views
Topology/geometry of $O(2n)/U(n)$
I am interested in what is known about the topology (diffeomorphism type) or geometry (is it complex? non-complex? symplectic?) of either the compact space of orthonormal complex structures on $\...
- 323
5
votes
1
answer
329
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Reference for the rectifiablity of the boundary hypersurface of convex open set
The boundary of any convex open set $X$ is $\mathbb R^n$ is a rectifiable hypersurface.
To see this, intuitively, simply take a sphere $S_d$ with diameter $d\in(0,+\infty]$ that contains $X$. The ...
- 263
13
votes
2
answers
2k
views
Is there a solution of the Yamabe problem using Ricci flow?
Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
- 5,146
4
votes
1
answer
317
views
Minimal graph over convex domain is area-minimizing
I am looking for a reference stating that
If a graph $z=f(x,y)$ over a convex domain $D$ is minimal, then it is area-minimizing.
5.4.18 in Federer's "Geometric measure theory" and Lemma 1.1. in ...
- 45k
5
votes
2
answers
2k
views
Canonical reference for Chern characteristic classes
I'm a little uncertain about the definitions for
Chern roots
Chern classes
Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
- 10.5k
14
votes
0
answers
312
views
An unpublished paper of Thurston about diffeomorphism groups
William Thurston has done many contributions in the field of diffeomorphism groups. But it seems that one of his papers entitled
"On the Structure of the Group of Volume Preserving Diffeomorphisms"
...
- 747
7
votes
1
answer
456
views
Nash embedding theorem for manifolds with boundary
A celebrated theorem of Nash is that every $C^k$ ($k\geq 3$) Riemannian manifold $(M,g)$ can be isometrically embedded into some Euclidean space $\mathbb{R}^d$ for some $d\in \mathbb{N}$. However, I ...
- 409
9
votes
2
answers
981
views
Text on old-fashioned differential geometry
I am seeking good books on the geometry of surfaces in Euclidean space, which would in particular discuss Darboux frames. Please explain for each suggestion why you like this book (classics are ...
- 14.4k
1
vote
0
answers
42
views
Effect of plumbing a surface on the marked length spectrum
First I'll recall the plumbing procedure.
Let $M$ be a noded Riemann surface with nodes $p_1,\dots, p_n$. There is a family of pairwise disjoint neighbourhoods of each node $U_i$ that has coordinates ...
- 445
5
votes
0
answers
272
views
When do surfaces in $\mathbb{R}^4$ intersect all their translations in one direction?
I am looking for research or references on the following problem.
Let $S$ be a smoothly embedded connected surface in $\mathbb{R}^4$, with or without boundary. Fix some axis in $\mathbb{R}^4$, let $d ...
- 1,763
5
votes
1
answer
953
views
Literature Request: Berger Spheres and their Construction
In a previous question by me I asked about Berger spheres and their Lorentzian analogue, squashed $AdS_3$ along Hopf fibres. It was well answered (by https://mathoverflow.net/users/13268/ben-mckay) ...
- 679
1
vote
1
answer
331
views
Can divergence free vector fields be approximated by smooth ones?
If $M$ is a compact Riemannian manifold, is the space of $C^{\infty}$ divergence-free vector fields dense in the space of $C^r$ divergence-free vector fields, in the $C^r$ topology ($r\geq 1$)? How ...
- 19
6
votes
0
answers
132
views
Generalization of pseudogroups
Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup
One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
- 181
7
votes
1
answer
483
views
Furthest distance half the diameter?
Let $S$ be the surface of a convex body, polyhedral or smooth,
embedded in $\mathbb{R}^3$.
For a point $x \in S$, let $F(x)$ be the set of furthest points
from $x$, measured by shortest paths on the ...
- 151k
17
votes
4
answers
2k
views
Differential geometry applied to biology
This was originally a question posted here on MathSE. But I'll ask again here to see if I can get some different answers.
I'm looking for current areas of research which apply techniques from ...
- 171
7
votes
2
answers
396
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 $g_{\epsilon}$ ...
- 6,741
4
votes
1
answer
727
views
Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds
On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
- 43
3
votes
0
answers
159
views
Upper bound on the geodesic distance in a Lipschitz domain
I was wondering if the following result is true. If yes, could you please suggest a reference. The result seems to have been used at several papers without quoting any reference. Is the proof ...
- 895
2
votes
0
answers
134
views
Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$
Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property:
For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
- 965
10
votes
1
answer
707
views
Injectivity radius of manifolds with boundary
This question stems from the discussion in:
how to define the injectivity radius of manifolds with boundary?
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. In this context, let ...
- 409
7
votes
1
answer
455
views
Properties of the affine curve-shortening flow
The affine curve-shortening flow (ACSF) seems to satisfy the following properties, even if the initial curve(s) is (are) self-intersecting (at least as long as they are "nice enough"):
(1) The length ...
- 1,174
1
vote
0
answers
159
views
The Laplacian of a tubular neighborhood
Let $(M,g_M)$ be a compact submanifold of $\mathbb{R}^n$. Are there any known results relating the spectrum of the Laplace-Beltrami operator of M to the spectrum of the Laplace-Beltrami operator of a ...
- 409
0
votes
1
answer
289
views
Estimate for Laplace equation with Neumann boundary on manifold with corner
Let $(M,g)$ be a compact Riemannian manifold with boundary and corner, i.e. locally modelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$.
...
- 1,915
5
votes
0
answers
261
views
The space of $k$ differential forms as a Fréchet space
Given a smooth manifold $M$, can define define seminorms on $\Gamma(U,\bigwedge^kT^{\ast}M)$ for $U$ a coordinate open set by the following: $p^{s}_L(u = \sum_{I}u_I dx_I) = \sup_{x \in M}\max_{|I|=p, ...
- 51
2
votes
1
answer
320
views
Flat scalar curvature on 4 manifold
Let $(M,g)$ be a closed(oriented) Riemannian 4 manifold. It is well-known that, if $scal^g\geq0$ and not identically zero, then $M$ admits a PSC metric by conformal transformation.
Q Is $T^4$ the ...
- 1,915
3
votes
0
answers
99
views
Partial regularity of harmonic maps into spheres
Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
- 1,407
7
votes
1
answer
296
views
ASD connection for Line bundle over $4$-manifold
Let $(M,g)$ be an oriented closed Riemannian $4$ manifold.
Let $L\to M$ be a complex line bundle.
Q Under what condition, we can find an ASD connection of $L$, i.e. a connection $A$ such that $F^+...
- 1,915
5
votes
1
answer
743
views
Eigenvalues and Domain of the Laplace-Beltrami Operator
Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
- 51
6
votes
0
answers
122
views
Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?
Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying:
$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$
Where $(r,\theta,\phi)$ ...
- 969
1
vote
0
answers
126
views
Paper "Tetrahedron rolled onto a plane"
I am looking for the article Charles W. Trigg, "Tetrahedron rolled onto a plane", J. Recreational Mathematics, 3(2):82–87, 1970.
It is from @Joseph O'Rourke comment in the previous post "Die-rolling ...
- 11
12
votes
1
answer
680
views
Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach
Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action.
There is an associated fibre bundle $E\rightarrow ...
- 2,792
3
votes
1
answer
286
views
Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein
Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein?
I was told that we can use the following method:
Step ...
- 79
14
votes
3
answers
3k
views
Errata for Bott and Tu's book "Differential Forms in Algebraic Topology"
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Tu is a prequel.
Is there a good list of errata for Bott and Tu available? ...
- 315
5
votes
1
answer
413
views
Casson invariant and Euler characteristic
A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference ...
- 31
14
votes
5
answers
1k
views
History of the notion of $(G,X)$-structure
I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...
- 403
0
votes
0
answers
64
views
Relationship between the vortex filament equation and the transport equation
Let us consider the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$.
How is the Cauchy problem for the ...
- 277
0
votes
1
answer
124
views
Relationship between the vortex filament equation and the cubic Schrödinger equation
How is the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
related to the cubic Schrödinger equation?
Note 1. ...
- 277
1
vote
0
answers
75
views
Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...
- 277