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2 votes
1 answer
350 views

If there exists a function on a Riemannian manifold such that its Hessian matrix is the identity matrix?

In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. While on a smooth manifold $(M^n, g)$, do there exists a function on $(M^n, g)...
Davidi Cone's user avatar
3 votes
1 answer
257 views

Space filling curves

The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$. It is ...
Rahul Sarkar's user avatar
4 votes
1 answer
236 views

What is the Freudenthal compactification of a wildly punctured n-sphere?

Let $C$ be a compact and totally-disconnected subspace of the $n$-sphere $\mathbb{S}^n$, where $n\geq 2$. Question: Must the Freudenthal compactification of $\mathbb{S}^n \setminus C$ be homeomorphic ...
Agelos's user avatar
  • 1,926
4 votes
0 answers
227 views

To what extent is the Nash embedding not unique?

Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique? It is clear that the set of all such embeddings ...
dennis's user avatar
  • 521
4 votes
0 answers
68 views

Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
Spencer Kraisler's user avatar
3 votes
0 answers
117 views

Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
Matheus Andrade's user avatar
3 votes
1 answer
154 views

Which quantity could hold its convergence under Gromov-Hausdorff convergence?

Recently I've been reading T.H.Colding's paper of Ricci curvature and volume convergence. A proof of the continuity of volume functions was given under the lower Ric bounded condition. Having searched ...
Fan Gauss's user avatar
2 votes
0 answers
71 views

Domain of definition of a certain mapping

Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold. I am studying the mapping $$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...
Giuseppe Tenaglia's user avatar
4 votes
0 answers
195 views

Classifying singularities of the Ricci flow

Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
Matheus Andrade's user avatar
13 votes
1 answer
654 views

Are there examples of Einstein manifolds with unbounded curvature?

Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$ I'm looking for these examples because they might provide a counter-example to a ...
Matheus Andrade's user avatar
0 votes
0 answers
134 views

Einstein submanifold of Einstein manifold - References

Is there any work in which the conditions under which an Einstein manifold admits an Einstein submanifold are studied? If yes, can you give me the references?
MathDG's user avatar
  • 272
7 votes
1 answer
546 views

Can a hyperbolic manifold be a product?

I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
Michael Albanese's user avatar
5 votes
1 answer
305 views

harmonic coordinates on non-compact manifolds

Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
Siran Victor Li's user avatar
2 votes
0 answers
95 views

Vector bundle endomorphism diffeomorphism invariant?

Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
phydev's user avatar
  • 91
11 votes
4 answers
2k views

Elliptic regularity on compact manifold without boundary

Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this: For any $u\in H^1(M)$, ...
S. Maths's user avatar
  • 571
1 vote
1 answer
423 views

Riemannian Manifolds of Bounded Curvature

I am a complete newbie Riemannian Geometry with a particular application in mind so please excuse a lack of rigor in the question. Suppose I have a manifold with sectional curvature everywhere ...
user1189053's user avatar
26 votes
2 answers
4k views

Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
trequartista's user avatar
16 votes
1 answer
599 views

If all balls around two points are isometric... -- manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin. Let's say that a metric space $(X,d)$ has two poles if: there are two distinct points $x$, $y$ such that ...
Marco Golla's user avatar
  • 10.9k
9 votes
1 answer
993 views

Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...
Felix Goldberg's user avatar
1 vote
1 answer
305 views

On the canonical neighborhoods

Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow and Geometrization of 3-Manifolds" book as a definition of canonical neighborhoods have ...
Sepideh Bakhoda's user avatar
34 votes
7 answers
16k views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
Sepideh Bakhoda's user avatar
7 votes
1 answer
1k views

About Sectional Curvature [closed]

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
Sepideh Bakhoda's user avatar
3 votes
1 answer
174 views

A k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of ...
Sepideh Bakhoda's user avatar
15 votes
3 answers
2k views

Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ ...
Renato G. Bettiol's user avatar
6 votes
2 answers
3k views

When a Riemannian manifold is of Hessian Typ

When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)
user avatar
7 votes
0 answers
669 views

Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between two Riemmanian manifolds $M_1$ and $M_2$ such that, for each $x_1 \in M_1$, the (codimension-1) measure of the set of points at distance $d$ from $x_1$ ...
Joseph O'Rourke's user avatar
0 votes
1 answer
271 views

Numbers associated with boundaries of manifolds

I don't know what name if any is attached to the numbers I'm about to describe. For a line segment, [a,b] the number is 1 if for any k in (a,b) and 2 if k=a or k=b. For a square, [a,b] ...
user6137's user avatar
  • 379
12 votes
1 answer
1k views

Riemannian metrics on non-paracompact manifolds

After proving the existence of Riemannian metrics on manifolds, one of the students asked if the "paracompactness" is necessary. Of course the standard proof with the partition of unity uses this ...
Keivan Karai's user avatar
  • 6,214
7 votes
2 answers
1k views

G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...
Dror Atariah's user avatar
14 votes
4 answers
6k views

When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
Theo Johnson-Freyd's user avatar