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.

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12
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1answer
533 views

Is there any progress on Problem 12 (from Schoen and Yau)?

I saw the following question from the "Problem Section" in Schoen and Yau, page 281, problem 12: Let $M_1, M_2$ each have negative curvature. If $\pi_1 (M_1)=\pi_1 (M_2)$, prove that $M_1$ is ...
13
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2answers
568 views

If there is a dense geodesic, are almost all geodesics equidistributed? Dense?

Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...
5
votes
1answer
256 views

The surjectivity of the exponential map for the isometry group

Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective. Let $M$ be a noncompact connected Riemann manifold, and ...
0
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0answers
84 views

Question about a particular estimate in Riemannian geometry

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
4
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1answer
71 views

The Chern connection on a Hermitian symmetric domain

There's a connection (the Chern connection) on the Tangent Bundle of a Kahler Manifold which is compatible with both the hermitan metric, and the holomorphic structure. In general, I guess there's no ...
0
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0answers
105 views

How to prove this inequality of heat flow from Weitzenbock formula?

Let $(M,g), (N,h)$be a compact Riemannian manifolds, $m:=\dim M, n:=\dim N\geq 2$, and $N$ is a non-positive curvature $K_N\leq 0$. All connections which appear below are the Levi-Civita connections. ...
19
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1answer
491 views

Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$. There are conditions on $\{ p_1, p_2, d \}$ for this ...
0
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0answers
66 views

The limit of a sequence of embedded minimal disks in $\mathbb{R}^3$

Let $\Sigma_n,n\ge 1$ be a sequence of embedded minimal disks in $\mathbb{R}^3$ such that: (1) $0\in\Sigma_n\subset B(0,r_n)$ with $r_n\to\infty$ as $n$ tend to $\infty$, (2) ...
2
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1answer
189 views

The points of half area of a triangle

Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...
0
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0answers
42 views

Quadric functions on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold such that the parallel transport along every simple closed curve is the identity operator. A smooth function $f:M\to \mathbb{R}$ is called a quadric function if ...
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0answers
81 views

The space of affine functions on a riemannian manifold

Let $(M,g)$ be a Riemannian manifold. A smooth function $f:M\to \mathbb{R}$ is called an affine function if its hessian $H(f)$is identically zero. The Hessian is defined as follows: ...
2
votes
1answer
119 views

totally geodesic submanifold of Heisenberg group

Let $G= \left\{ \begin{pmatrix} 1&a&c\\0&1&b\\0&0&0 \end{pmatrix} \mid a,b,c\in \mathbb{R} \right\}$ be the Heisenberg group. Is there a compact codimension one submanifold ...
2
votes
1answer
177 views

What is the difference between $\delta W^{\pm}=0$ and Einstein?

Maybe this is a vague question. In Besse's book Einstein manifolds, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference ...
9
votes
1answer
231 views

Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a ...
4
votes
1answer
160 views

Analytic representatives for Kahler classes

If we are given compact complex manifold $X$ and a Kahler class $[\omega]$, can we always find a positive definite representative $\omega \in [\omega]$ that is real analytic?
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0answers
187 views

The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
3
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0answers
52 views

Hessian eigenspaces form integrable distributions on a Riemannian manifold?

Suppose $M$ is a Riemannian manifold and $f:M\to\mathbb{R}$ a differentiable function. I can form the Hessian $H$ of $f$ (with respect to the Levi-Civita connection); this is a symmetric bilinear ...
2
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0answers
107 views

On the volume entropy of negatively curved manifolds

Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the ...
3
votes
3answers
215 views

Is the group of isometries of a homogeneous Riemannian manifold maximal?

I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that: Iso is a proper subgroup of ...
2
votes
0answers
104 views

Variational inequality on Manifold

Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla ...
1
vote
3answers
119 views

Harmonic Function with special property

I would appreciate any help with the following problem: Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...
2
votes
1answer
256 views

A question on Schrodinger operator

I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated! Let $(M,g)$ be ...
1
vote
1answer
104 views

Volume bounds of balls in Riemannian manifolds

Let $(M,g)$ be a complete Riemannian manifold and suppose $\mathrm{Ric}(g) \geq -k$ for some $k>0$. Suppose we know that $\mathrm{vol}_g (B_1^g (x_0)) \geq \nu$ for some particular $x_0 \in M$ and ...
4
votes
1answer
209 views

Volume of geodesic balls

I have two questions (somewhat related) regarding local geometry on a SMOOTH, COMPACT Riemannian manifold. I still have a hard time getting a "good" understanding of local geometry. Question 1: It ...
2
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0answers
109 views

Counterexample to volume comparison inequality assuming only scalar curvature bound?

The Gromov-Bishop volume comparison theorem says that if we have a lower bound for the Ricci curvature on $(M,g)$, then its geodesic ball has volume not greater than the geodesic ball with the same ...
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0answers
49 views

Coarse geometry of minimal surfaces in non-positively curved manifolds

Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface. (You can basically image $X$ as a ball and $S$ as an embedded disk whose ...
3
votes
0answers
110 views

$\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$. Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...
-4
votes
1answer
119 views

compact complex manifolds and complet curves [closed]

let $X$ be a compact complex manifold of dimension one . my first question is : 1) -does all compact complex manifolds of dimension one admit nonconstant meromorphic function ? . now , let ...
2
votes
1answer
190 views

Taylor expansion of the determinant of a Riemannian metric

Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ ...
1
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0answers
76 views

A question on the maximum principle for Schrodinger operators

Let $(M^n,g)$ be a closed Riemannian manifold, and $L=-\Delta+V$ be a Schrodinger operator, $V\in C^{\infty}(M)$. In answers to the two questions (First eigenvalue of Schrödinger operator is simple 1 ...
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0answers
98 views

Estimate the smallest eigenvalue of a Schrodinger operator

There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of Grigor'yan-Nadirashvili-Sire and references therein. I wonder how to ...
0
votes
1answer
183 views

Describe all differentiable functions on $\mathbb{S}^n \backslash S$ (S is the south pole) [closed]

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
2
votes
1answer
119 views

local approximation of a vector field on a Riemannian manifold

Let $(M^n,g)$ be a Riemannian manifold, and let $V$ be a $C^{\infty}$ vector field on $M$. Is it possible to locally approximate $V$ by gradient vector fields $\nabla f_i$, such that the ...
0
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0answers
54 views

Local behavior of Killing spinor on Sasaki-Einstein Manifold

I am trying to understand how a Killing spinor behaves near a closed Reeb orbit, for instance, on $S^5$ and $Y_{p,q}$ manifolds So Let us consider the Killing spinor equation on a five-dimensional ...
1
vote
1answer
227 views

Sharp Gaussian upper bounds on Heat Kernel

I am looking for references (with proof) for the following statement: Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be ...
0
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0answers
82 views

Application of conformal normal coordinates for higher order elliptic operator

Let $n>2$ be even. Consider a compact Riemannian manifold $(M^n,g)$ and denote with $P_g$ the critical GJMS operator. Recall that $P_g$ is conformally invariant, i.e. $$P_{\tilde g}=e^{-nu}P_g$$ ...
1
vote
1answer
150 views

de Rahm Laplace operator on forms bounded

Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The ...
3
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0answers
162 views

Can we obtain topology results using analysis in metric measures spaces?

Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I ...
1
vote
2answers
334 views

Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$, i.e., each $p=(p(x_0),\dots,p(x_n))\in ...
1
vote
1answer
123 views

Is the structure constant additive on connected components?

Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its ...
0
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0answers
73 views

Decomposition of the canonical flat connection on $\tilde M\times_{\rho} SL(n,\mathbb{C})$

I'm looking for a proof resp. reference for a statement of the following form: Let $M$ be a compact Riemann surface, $\tilde M$ its universal covering, $\rho$ a semisimple representation of its ...
1
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0answers
48 views

Bounds on functions pullbacked via exponential map

Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting $$ (\exp_x^* ...
0
votes
1answer
74 views

Optimal Regularity for Invariance of Curvature under Isometries

It is well known that sectional curvature is an invariant under isometries. I wonder what the optimal regularity for this result to hold is (in terms of Hölder-spaces)?. It is classical that ...
12
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2answers
856 views

Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?

I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...
3
votes
1answer
246 views

Special Riemannian connections?

Assume that $E$ is a bundle of Lie Algebras. Let $g$ be an invariant metric on $E$, that is for all $p\in M$, $$g_p([x,y],z)+g_p(y,[x,z])=0,$$ where $x,y,z\in E_p$ are arbitrary. Is there a ...
7
votes
2answers
403 views

Obtain Lorentzian manifolds from Riemannian ones by Wick rotation

In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian ...
2
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0answers
98 views

Notions of convexity for the boundary of a Riemannian manifold

Let $(M,g)$ be a compact smooth Riemannian manifold with boundary $\partial M\subset M$. Question: What does it mean to say that the boundary is convex and strictly convex? Alternatively: What ...
1
vote
1answer
142 views

Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
4
votes
1answer
98 views

Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function, ...
1
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1answer
202 views

A special type of transitivity

Let $M$ be a smooth orientable manifold with volume form $\Omega$. Fix two pints $x,y \in M$. Put $A$=all volume preserving diffeomorphism of M which maps $x$ to $y$. Define $B$=All linear volume ...