Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
81 views

What is wrong with the derivation?

Let $(M^n,g)$ be a Riemannian manifold, and $T$ a symmetric $(1,1)$-tensor field, i.e., $\langle T(X),Y\rangle = \langle X,T(Y)\rangle $. For convenience, denote $$\Delta_Tu=\sum_i\langle \nabla_{e_i}\...
2
votes
0answers
88 views

If the total space of circle bundle over hyperbolic manifold admits Riemannain metric of non-positive sectional curvature?

If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature? I wish to use the result about the question and find Leeb's work 3-...
3
votes
0answers
97 views

Flat solvmanifolds?

I am trying to look for some reference for solvmanifolds and come up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat ...
2
votes
0answers
104 views

Do we have estimate like $\int_\gamma \alpha \le |\alpha| \cdot |\gamma|$?

Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing $$ H^1(X) \times \pi_1(X) \to \...
0
votes
1answer
86 views

Differentiating Riemannian logarithmic map

Let $(M,g)$ be a Riemannian manifold, geodesically complete, and assume logarithms are well defined and smooth. Let $c: I\to M $ be a smooth path in $M$, and $x\in M$. Can we say something about $$\...
11
votes
4answers
320 views

Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...
4
votes
1answer
84 views

Conformally flat homogeneous spaces

Let's say we have a homogeneous space $H\backslash G$. Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure? I am particularly ...
1
vote
0answers
36 views

Diffusion generators with gradient vector fields

Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as $$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$ where $X_0,X_1,...,X_k$ are ...
4
votes
3answers
253 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 $u\in H^1(M)$, ...
1
vote
1answer
78 views

Is there Brownian motion on Alexandrov spaces?

It is well known that there is a notion of Brownian motion on smooth Riemannian manifolds. I am wondering if there is a more general notion of Brownian motion on finite dimensional Alexandrov ...
6
votes
1answer
247 views

Simple application of Bochner--Weitzenböck type formulas

I am looking for a cool and simple (but not worn-out) application of Bochner--Weitzenböck type formulas which could be used as a motivation. What is your favourite example? P.S. Here is one which ...
2
votes
0answers
125 views

Existence of a certain kind of compact spin manifold with boundary

For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
0
votes
0answers
78 views

Vector-valued forms in Riemannian geometry

Suppose $(M,g)$ is a Riemannian manifold. I want to find a vector-valued $2-$form $T$ such that, for any vector fields $X,Y,Z$ on $M$, $$ g(T(X,Y),Z)=g(T(Z,X),Y)\,. $$ As a motivation, consider the ...
7
votes
1answer
163 views

The developing map of conformally flat manifold

There is one sentence I don't understand in some paper. "A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map. Is any ...
7
votes
2answers
300 views

Symplectic connections are (locally) Levi-Civita connections

I was wondering... Is every symplectic connection $\nabla$ on some symplectic manifold $(M,ω)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?
7
votes
3answers
322 views

Sectional curvature of leaves of foliation

Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ ...
9
votes
2answers
335 views

Deforming metrics from non-negative to positive Ricci curvature

Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one? I know it's impossible in general due to ...
21
votes
7answers
2k views

Smooth functions on sphere

Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...
5
votes
1answer
132 views

Killing vector fields of a conformally flat Riemannian metric

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a smooth function and let's consider the conformally flat Riemannian metric $g = e^f \delta_{ij} dx^idx^j$ on $\mathbb{R}^n$. Is it true that the Killing ...
4
votes
0answers
109 views

Trace free Codazzi Tensor on Hyperbolic manifolds

Does there exist trace free nontrivial symmetric Codazzi Tensor on closed manifold with constant sectional curvature -1? I know, locally all Codazzi Tensors on closed manifold $(M, g)$ with constant ...
3
votes
0answers
92 views

An upper bound for the number of singularities of a transversal vector field isometric to the zero field

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$. A smooth vector field $X:M \to TM$ is called a transversal vector field if $X(M)$ is transverse ...
10
votes
1answer
231 views

Positive Ricci curvature on fiber bundles

My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture: Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...
0
votes
0answers
67 views

Eigenfunctions of square root of Laplacian in an arbitrary Riemannian manifold?

Please forgive me for my inability to pose a mathematics question properly. In one dimension the eigenfunctions of Laplacian (simply double derivative) are also eigenfunctions of its square root (...
1
vote
0answers
111 views

The converse to the positive mass theorem

Let $(M^n,g)$ be an asymptotically flat manifold of decaying-order $\tau>\frac{n-2}{2}$, the positive mass theorem states that if the scalar curvature $S_g$ is non-negative, then the ADM mass $m_g$...
3
votes
2answers
172 views

Einstein warped product manifold Ricci flat

Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold Ricci flat (i.e. $Ric=\lambda g$ with $\lambda=0$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and ...
6
votes
2answers
169 views

Generalization of Bieberbach's second theorem

Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...
1
vote
1answer
97 views

Exponential decay of resolvent kernel

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
4
votes
0answers
107 views

Orthonormal vector fields on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$. Assume that $X: M \to TM$ is a vector field on $M$. We say that $X$ is an orthonormal vector field ...
4
votes
0answers
47 views

Lichnerowicz-Obata theorem for symmetric 2-tensor

Let $(S^n/\Gamma,g)$ be the standard space-form with constant sectional curvature, where $\Gamma \subset O(n+1)$ is a finite group. If there exists a nonzero transverse-traceless symmetric 2-tensor $h$...
5
votes
1answer
108 views

The Hausdorff dimension of the union of singular orbits and exceptional orbits

Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
8
votes
1answer
232 views

Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones

Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...
1
vote
1answer
236 views

Gaussian curvature of a surface does not take the constant value 1?

I came across this very complex equation (calculating the Gaussian curvature of a surface): \begin{align*} 1 \not\equiv &-\frac{m}{2}\Bigl(\frac{3}{2}C+Su^{-1}-Tu^{-1}+2+Qu^{-1}\Bigr)\\ &\...
10
votes
2answers
345 views

Can scalar curvature and diameter control volume?

Scalar curvature can control the volume of geodesic ball locally, however, it can not bound the diameter. As far as I know, the example for a manifold with a large scalar curvature and volume has ...
6
votes
0answers
95 views

Does an 8 dimensional compact Riemannian manifold contain an embedded minimal hypersurface?

It is well know that if $(M^{n+1},g)$ is a compact Riemannian manifold and $n \leq 6$ then there exists a smooth, embedded minimal hypersurface $\Sigma^n$ in $M$ (infinitely many such $\Sigma$, even) ...
2
votes
0answers
130 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 ...
1
vote
0answers
49 views

Second variation in saddle Finsler surface

Setting : Consider a two dimensional surface in $ (\mathbb{R}^n,\|\ \|)$. Here we define a function $f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ s.t. $L(v)(X)=\langle f(v),X\rangle$ where $\langle\ ,\...
1
vote
1answer
117 views

Convexity of curves in Riemannian surfaces

It is known that a curve $f:[0,2\pi]\to \mathbf{R}^2$ is convex if $\partial_t (\arg f'(t))\ge 0$. My question is: does this statement have an analogue in the setting of Riemannian surfaces instead of ...
2
votes
0answers
74 views

geodesic balls in the conformal change

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...
3
votes
1answer
91 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 ...
8
votes
2answers
298 views

When $\int_M \exp(-d_M(x,y)^2/t) dvol(y)$ becomes constant for a Riemannian manifold $M$?

Let $(M,g)$ be a closed and connected Riemannian manifold. $d_M$ is its geodesic metric and $dvol_M$ is its standard volume measure. For each $t>0$, define a map $f:M\rightarrow\mathbb{R}_{>0}$ ...
7
votes
1answer
214 views

Log-concavity of areas of level sets

Suppose $f: \mathbb{R}^d \to \mathbb{R}$ is a smooth convex function. Consider the level sets of the function, namely $M_s = \{x: f(x) = s\}$. Is it true/known that the surface areas of $M_s$ are ...
1
vote
0answers
81 views

Solutions to $\Delta u\ge u^2$

Let $(M,g)$ be a complete Riemannian manifold. Suppose that $u$ is a nonnegative solution to $\Delta_gu\ge u^2$. Does it follow that $u$ must be identically 0? I know that the answer to above ...
1
vote
0answers
60 views

Some hypersurface has a positive second fundamental form potentially

Notation : $r^2=x^2+y^2$. Exercise : Define $$F_\sigma (x,y)= (f_\sigma (x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2 r^2)(x-\sigma x^3,y-\sigma y^3)$$ Define $ G_\sigma: \mathbb{...
6
votes
0answers
79 views

Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9?

In page 15 of the article New Applications of Min-max Theory, Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi_{1,2}$ in the 3-sphere to have a Morse index 9, ...
1
vote
0answers
86 views

Weak elliptic maximum principle on manifolds without strict ellipticity

This question is not to be confused with the similarly titled question here. In the above lined question, I gave a complete answer, but noticed that things are apparently not so simple in the ...
5
votes
1answer
223 views

Moduli space of flat connections over a Riemann surface

If I understand correctly, in the Refs below: We can see that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\...
6
votes
0answers
93 views

Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?

QUESTION: Let $n$ be a natural number. Is it true that there exist $N(n), D(n) > 0$ such that any complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $N$...
13
votes
0answers
167 views

Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?

Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...
13
votes
0answers
198 views

Lorentzian analogue to Thurston geometries

Is there an analogue to the eight Thurston geometries for Lorentz metrics? If so, how many "disctinct" geometries are there in the Lorentzian case? And which closed 3-manifolds admit metrics which ...
3
votes
0answers
69 views

Classification of Euclidian-like Klein geometries in spirit of Erlangen program

All we know about Erlangen Program approach to geometry. The main idea is to consider Lie Group $G$ acting transitively on smooth manifold $M$ (e.g. nLab). I really like such style of thinking, but ...