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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Solution to Seiberg-Witten monopole equation

To understand some physics problem, I want to know if there is (non-$L^2$ or $L^2$) solution to the Seiberg-Witten equation on $\mathbb{R}^4$ \begin{equation} {D}_A \psi = 0\\ F_A^+ = ...
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389 views

Ricci curvature under rough convergence

From the work of Lott--Villani and Sturm, I know that the following fact holds: (*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci ...
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608 views

Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$: \begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation} This equation defines a foliation on ...
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3answers
491 views

Elliptic theory on compact manifolds

Maybe this is silly. On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation $$ \Delta u=f \quad\text{ in $\Omega$}$$ $$ u=0\quad\text{ on $\partial\Omega$}.$$ One has the following ...
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Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
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Riemann normal coordiantes and change of metric

Le $(M,g)$ be Riemannian manifold. Fix point $p\in M$. We can define the map $$\exp: U \subset T_p M \rightarrow M$$ $$\exp(X) = \gamma_{p,X}(1)$$ where $t\mapsto \gamma_{p,X}(t)$ is geodesic such ...
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300 views

Two questions on isometric embedding

According to the answer of the following question, I try a new version: An special isometric embedding Let $M$ be a Riemannian manifold and $\gamma$ be a small part of a geodesic. Is there an ...
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85 views

An special isometric embedding

Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic. Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line? The second ...
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562 views

The trace of a wedge product of matrices

I'm trying understand a computation on page 371 of Besse's book on Einstein Manifolds. I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form ...
2
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1answer
107 views

Pullback of $L^p$ functions via exponential map

Let $M$ be a complete Riemannian manifold, endowed with its exponential map $\exp: TM \longrightarrow M$. For any $C^k$- function $u$, we get the Pullback $$ \exp^* u = u \circ \exp$$ which is in ...
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tangent developable surface in $\mathbb{R}^3$

Let $C$ be a regular curve embedded in $\mathbb{R}^3$ (i.e. a real 1-dimensional manifold embedded in $\mathbb{R}^3$). Let $S$ be the union of its affine tangent lines: $$S=\bigcup\limits_{p\in ...
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“Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
2
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1answer
199 views

Killing constant in Killing spinor equation

This is a simple question, but I can't find explicit discussion in literatures that I can find. Real/imaginary Killing spinor equation \begin{equation} \nabla_\mu \psi = \lambda \gamma_\mu \psi ...
2
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1answer
275 views

Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$?

Define $$Q = \bigcup_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required. Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
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1answer
154 views

Hamilton's Proof of the Tensor Maximum Principle

My questions come from Richard Hamilton's Three-Manifolds with Positive Ricci Curvature paper. I'm trying to work through parts of the paper so I can better understand the Ricci Flow for my research. ...
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504 views

Second derivative of Riemannian Exponential Map

Let $M$ be a Riemannian manifold. Let us look at the Riemannian exponential function $\exp_x: T_x M \supset \mathcal{D} \longrightarrow M$. The derivative of the exponential map can be expressed in ...
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221 views

Brakke's theorem for gap in entropy between self-shrinkers

In their paper The round sphere minimizes entropy among closed self shrinkers, Colding-Ilmanen-Minicozzi-White state "It follows from Brakke's theorem that $\mathbb{R}^n$ has the least entropy of any ...
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2answers
546 views

A Converse to the Gauss Bonnet Theorem

Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent ...
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2answers
166 views

Normal vector field associated to deformations of Riemannian submanifolds

Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $X$ be a immersed submanifold in $M$ of dimension $k$ i.e there is a immersion $F_{0}:X \longrightarrow M$. A deformation of the submanifold ...
2
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0answers
109 views

How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper. Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ On page 5 of ...
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1answer
283 views

Monograph or rich survey on infinite-dimensional Riemann manifolds

I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...
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1answer
77 views

'Convex' slices of proper actions

Consider a Lie group $G$ acting properly on a manifold $M$. Then by the slice theorem we can find for any point $m\in M$ a submanifold transverse to the orbit $\mathcal{O}$ through $m$ and which is ...
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341 views

Reference request for an early theorem of Gromov

In his talk Misha Gromov- How does he do it, Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: Every manifold admitting a sequence of metrics such that the diameter ...
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4answers
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What is the analog of the “Fundamental Theorem of Space Curves,” for surfaces, and beyond?

The "Fundamental Theorem of Space Curves" (Wikipedia link; MathWorld link) states that there is a unique (up to congruence) curve in space that simultaneously realizes given continuous curvature ...
4
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1answer
255 views

Intersections of complex submanifolds in $\mathbb{C}^N$

This is an exercise from Gromov's Partial differential relations. (page 5) Let $V$ and $V'$ be two closed complex submanifolds in $\mathbb{C}^N$ of complimentory dimension. Prove that $V$ and $V'$ ...
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1answer
635 views

What is the geometric interpretation of this quantity?

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic structure on the tangent bundle, $(TM, ...
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1answer
119 views

Compact Riemannian manifold with maximum average distance

Given a compact connected Riemannian $2$-manifold $M$ with positive curvature (thus by Gauß-Bonnet, $M$ is diffeomorphic to a 2-sphere) and diameter 1, what is the supremum (as $M$ varies over all ...
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2answers
223 views

Some facts about cut-locus

Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic to $S^2$. S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree." How the set of point can be a tree? ...
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0answers
77 views

Deduce global estimate from scaling-invariant local estimate

Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
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996 views

Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result: The geodesic flow on a manifold with negative curvature is ergodic. The lecture note that ...
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1answer
162 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
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3answers
465 views

Number of disjoint simple closed geodesics

According to Jairo comment on the first version of this question I revise the question as follows; Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that: There are at most a ...
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0answers
195 views

Surface with bounded torsion propery [closed]

Let $S$ be a surface in $\mathbb{R}^{3}$ with the following property: There is a uniform constant $M$ such that for every Frenet curve $\gamma(t)$, contained in $S$, $| \tau(t) | \leq M$, for all ...
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Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic n-space. Let $TH$ be the tangent bundle of $H$, endowed with its Sasaki metric. I have two questions: Is $TH$ isometric to $H$ times a flat n-space? What is the group of ...
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1answer
242 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
9
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1answer
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Spectrum of Laplacian in non-compact manifolds

What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty? What would be a ...
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Smoothing of the distance function on a Riemannian manifold

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let ...
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1answer
158 views

Derivative of (the length of) the Ricci tensor

I was wondering, have you ever seen a formula in the Riemannian (more specially Kahlerian but not essential) setting for the derivative $X \cdot |Ric|^2 = 2 g(\nabla_X Ric, Ric)$ for a vector field ...
5
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3answers
324 views

Is the set of focal points of a submanifold on a normal geodesic discrete?

Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...
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Can a manifold have a curvature-free connection that is not torsion-free?

Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a ...
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564 views

Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...
4
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2answers
201 views

Are ramified covering of negatively curved manifolds negatively curved?

Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, ...
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Existence of harmonic maps between loops

Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy ...
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1answer
43 views

Prescribing finitely many unparameterised planar geodesics

Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing ...
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2answers
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On the definition of convergence of a sequence of sections of a bundle

Convergence of a sequence of sections of a bundle is defined as follows: Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on ...
5
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1answer
206 views

Can an open manifold with positive Ricci curvature be non simply connected at infinity?

The question is in the title, I haven't been able to locate a discussion of these kind of properties.
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1answer
287 views

Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
11
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3answers
265 views

Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?

The geometrization theorem tells us: Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a ...
2
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1answer
233 views

Curve on a surface defined by its geodesic curvature

Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...
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1answer
350 views

Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...