Questions tagged [riemannian-geometry]
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
3,083 questions
20
votes
0
answers
2k
views
Schoen and Yau's proof of the higher dimensional positive mass theorem
In April 2017 Schoen and Yau posted on the arxiv their solution of the time-symmetric positive mass theorem in all dimensions, which has been a significant conjecture since the 70s. As of now, July ...
3
votes
1
answer
292
views
Identification of tangent spaces by parallel transport along geodesics [closed]
Given a geodesically complete manifold M, can we define a global identification of tangent spaces by starting from a base point, and parallel transporting along smooth geodesics? For this to be ...
- 12.3k
0
votes
0
answers
198
views
Metric in $T^*M$ with positive injectivity radius and with totally geodesic fibers
Suppose we have a cotangent bundle $T^*M$ of a compact manifold $M$. If we would consider the Sasaki-metric on $T^*M$ we would be able to find a metric which has totally geodesic fibers.
Now I also ...
- 791
4
votes
1
answer
230
views
Generalizing a result about hyperbolic 2-folds to hyperbolic 3-folds
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ \Sigma_g $ be a compact orientable surface of genus $ g $. Let the subgroup $ \pi_1(\Sigma) $ of $ \SL_2(\...
- 10.4k
4
votes
1
answer
134
views
Rigidity of the compact irreducible symmetric space
Let $(M^n,g)$ be an irreducible symmetric space of compact type. In particular, $(M^n,g)$ is an Einstein manifold with a positive Einstein constant.
Is there any classification for $(M^n,g)$ if $(M^n,...
- 12.9k
4
votes
0
answers
274
views
How many ways are there to characterise $\mathbb{P}^n$?
Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\...
- 1,379
3
votes
0
answers
268
views
The double exponential map and the Baker–Campbell–Hausdorff–Dynkin series
$\DeclareMathOperator\Exp{Exp}$A. Gavrilov has several nice works studying the double exponential map and its properties ([1] and references therein).
Given a complete Riemannian manifold $(M,g)$ and ...
- 31
5
votes
0
answers
180
views
Curvature of the line bundle $\mathcal{O}(2)$ on the twistor space
Let $M^4$ be a closed Riemannian manifold and $Z:=S\big(\Lambda^2_+(M)\big)$ denote the twistor space of $M,$ i.e., the sphere bundle of the self-dual 2-forms on $M$. Now at a point $(m,J)\in Z$ the ...
- 26.3k
9
votes
0
answers
336
views
Nash embedding for 3 manifolds
The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$ (edit: 14 is a better bound for compact 3 manifolds thanks @mme). What ...
- 4,081
4
votes
1
answer
245
views
Tzitzeica surface
A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface ...
- 108k
13
votes
2
answers
1k
views
What is known about Lie groups with (strictly) positive curvature?
If we consider $G$ a compact Lie group, there is a left invariant Riemannian metric whose the sectional curvature is nonnegative (see Milnors' paper). When can we find a left invariant metric that has ...
- 245
5
votes
1
answer
339
views
Finding vector fields on $S^2$ with equal divergence
Let $\mathfrak{X}_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}_{CK}$ be the 6-dimensional space of conformal ...
- 7,197
1
vote
0
answers
259
views
Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...
- 969
4
votes
1
answer
149
views
Is a linear vector field a geodesible vector field?
I have already asked this question in MSE; I repeat it here at MO.
Assume that $A\in M_n(\mathbb{R})$ is a non-singular matrix.
Is the flow of linear vector field $X'=AX$ a geodesible flow on $\...
- 9,237
3
votes
0
answers
216
views
Is there a measure for the space of submanifolds?
Let $(M,\mu)$ be a pair of a manifold $M$ ($C^\infty$ or Riemann if you like) and a probability measure $\mu$ on $M$. Is there a sensible way to put a probability measure on the space of submanifolds ...
- 549
1
vote
0
answers
172
views
Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563
3
votes
1
answer
470
views
Is the Moebius strip Riemannian homogeneous?
Let $ M $ be the Moebius band. In other words, the total space of the nontrivial line bundle over the circle. Can we equip $ M $ with a metric such the the isometry group acts transitively?
My ...
- 4,081
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
5
votes
1
answer
184
views
Proof of equivalence between Lie triple systems and totally geodesic submanifolds
In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a ...
- 13.5k
2
votes
0
answers
127
views
Are metrics of the form $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat?
Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$.
Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric ...
- 179
15
votes
1
answer
1k
views
Ricci curvature : beyond heat-like flows
Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs.
There are at least two versions of Ricci curvature in the ...
- 2,247
3
votes
0
answers
161
views
How to show the upperbound of the Ricci tensor preserved on 3 manifold
So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of ...
- 273
6
votes
1
answer
148
views
Isometric imbedding of a 2-disk into Euclidean 3-space
Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
- 45k
3
votes
0
answers
79
views
Virtually abelian fundamental groups equivalent to nonnegative curvature
This is a follow up question inspired by
Fundamental groups of compact manifolds with non-negative Ricci curvature.
In dimensions 3 and 2 (and 1) a manifold has a virtually abelian fundamental group ...
- 4,081
3
votes
1
answer
118
views
Is a cap an Alexandrov space?
Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
- 45k
5
votes
1
answer
228
views
An example of an SRB measure which is not a physical measure
Let $f:M \to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold with a normalized Riemannian volume $\mathrm{Leb}$. Given an $f$-invariant Borel probability $\mu$ in $M$, we call the ...
- 3,928
4
votes
0
answers
862
views
History of Laplacian comparison theorem
The Laplacian comparison theorem says that if a $n$-dimensional Riemannian manifold has nonnegative Ricci curvature, then the distance function to any point satisfies $\Delta d\leq\frac{n-1}{d}$. ...
- 12.9k
7
votes
2
answers
499
views
Submanifolds of Lie groups with abelian normal bundle
Let $M$ be a submanifold of a symmetric space $Q$. The normal bundle $NM$ is called abelian if $\exp(N_{p}M)$ is contained in some totally geodesic and flat submanifold of $Q$ for all $p \in M$; see ...
- 987
2
votes
0
answers
101
views
Parallelism defect
I have a question that I don't know how to answer.
If I have a parallelism defect it is due to the presence of a curvature and therefore we can bring it back to a Riemann tensor.
The thing that is not ...
- 63.9k
3
votes
0
answers
68
views
Ricci deformation of hyperkahler ALE orbifold
Let $(X^2,g)$ be a hyperkahler ALE orbifold surface. Consider its Ricci deformation equation:
$$
\Delta h+2Rm(h)=0
$$
for $\text{div}_g h=\text{Tr}_gh=0$ and $h=O(r^{-\epsilon})$ as $r \to +\infty$. ...
- 891
5
votes
0
answers
101
views
When are nodal lines on a sphere geodesics?
Let $(S^2, g)$ be a Riemannian sphere and let $L := \Delta_{S^2} + q$ be a Schrödinger operator on $S^2$. Suppose that $L$ has index equal to one and that $u \in C^{\infty}(S^2)$ ($u \neq 0$) lies in ...
- 2,095
6
votes
3
answers
368
views
Curvature function as a random variable with uniform distribution
Let $(S,g)$ be a Riemannian surface. Then the curvature function $\kappa: S\to \mathbb{R}$ can be counted as a random variable. So it produces a probability density function
$f_g:\mathbb{R}\to \...
- 52.3k
2
votes
1
answer
118
views
Can we always find coordinates on a surface such that $K=K(u-v)$?
Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function ...
- 91.8k
5
votes
0
answers
261
views
Laplacian spectrum and measured Gromov-Hausdorff convergence of Riemannian manifolds with boundary
In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured ...
- 409
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
2
votes
0
answers
78
views
Examples of curvature-adapted subgroups of semi-Riemannian groups
Let $G$ be a semi-Riemannian group, i.e., a Lie group equipped with a bi-invariant semi-Riemannian metric. I am looking for examples of Lie subgroups that are curvature adapted to $G$.
First, allow me ...
- 987
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?
$$X_0\...
- 28.2k
8
votes
1
answer
433
views
What should a meaningful notion of curvature satisfy, in the absence of a smooth structure?
There are many generalizations of various curvatures to non-smooth metric spaces (e.g. Ollivier's Ricci curvature). Suppose I have a metric space $(X,d)$ and I want to define a notion of curvature ...
- 6,001
2
votes
1
answer
257
views
A question about Dirac operators
Let $D$ be a Dirac operator on spinor bundle $S$ over even-dimensional non-compact spin manifold $X$,
$$
\left<s_1,s_2\right>_{L_2}
= \int_X \left<s_1,s_2\right> \quad \forall s_1,s_2\in\...
- 20.9k
2
votes
0
answers
149
views
Comparison of sum of vectors and exponential map on a Riemannian manifold
Suppose $M$ is a simply-connected complete Riemannian manifold with bounded sectional curvature $\delta \leq K \leq \Delta < 0$. Let $p_0\in M$. Define the sequence of points $p_1, \ldots, p_n$ by
$...
- 63.9k
5
votes
1
answer
194
views
3-manifolds with all minimal surfaces closed
Question. Let the manifold $(M^3,g)$ be compact without boundary. Suppose that every complete, embedded minimal surface $\Sigma \subset M^3$ is closed. Must $M$ be diffeomorphic to $\mathbf{S}^3$ or $\...
- 28.2k
3
votes
1
answer
327
views
Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$?
Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$ and geometrically which things it represents?
- 131
1
vote
0
answers
217
views
Research topics about non Euclidian geometry? [closed]
The title pretty much sums it up. I am wondering if anyone has some interesting research topics to investigate about non-euclidian geometry?
Sorry for the confusion, I am a first year undergrad, so ...
1
vote
0
answers
73
views
Relationship with between Clifford multiplication and pullback
Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
- 563
10
votes
0
answers
192
views
Metrization of projective manifolds
A modern take on Hilbert's fourth problem could be as follows:
Given a manifold $M$ with a flat projective structure (i.e., a $(PGL(n+1),\mathbb{RP}^n)$-structure), find all metrics for which the ...
- 13.5k
1
vote
1
answer
75
views
A property for generic pairs of functions and metrics
Let $M$ be a compact smooth manifold with a smooth boundary. Given a smooth Riemannian metric $g$ on $M$, we denote by $\{\phi_k\}_{k=1}^{\infty}$ an $L^2(M)$--orthonormal basis consisting of ...
- 5,048
6
votes
1
answer
816
views
Proof that every three-dimensional Einstein manifold has constant curvature
In pseudo-Riemannian geometry it is well known that every three-dimensional Einstein manifold has constant curvature. A proof of this is sketched here.
Question. Does anyone know where in the ...
- 7,127
3
votes
2
answers
347
views
Direct calculation of the Fisher distance via Riemannian geodesics
I'm looking for a reference for a direct calculation of the Fisher distance (to avoid overloading the term "metric") $d_F(x,y) := 2 \cos^{-1} \sum_i \sqrt{x_i y_i}$ as the geodesic distance ...
- 15.4k
1
vote
0
answers
141
views
Injecitivity radius of Sasaki metric
Suppose we have a compact riemannian manifold $(M,g)$ and we endow $TM$ with the Sasaki metric $\tilde g$. Now I am interested in understanding the injectivity radius of $(TM,\tilde g)$ but I am ...
- 791
3
votes
0
answers
397
views
Differential of exponential map with respect to the base point
Let $(M,g)$ be a smooth Riemannian manifold embedded in $\mathbb{R}^m$. I would like to understand the transformation formula which will allow me to pass from the integral $\int_M \dots dV_g(x)$ to $\...
- 319