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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5
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185 views

How to define the square root of $1-\Delta $?

If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$? The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential ...
2
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1answer
109 views

Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left ...
5
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5answers
822 views

Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?

Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? I am especially interset in the case ...
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1answer
116 views

Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question We consider the following two classes of smooth maps on ...
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0answers
32 views

Bounded Ricci curvature implies bound on Jacobi determinant?

Assume that $M$ is a complete Riemannian manifold and define for $X \in T_x M$ $$j(X) = \bigl|\det d \exp_x|_X \bigr|.$$ This is a smooth function on $TM$ for $X$ close enough to the zero section. ...
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0answers
56 views

Fill radius and fundamental group

I am reading M. Ramachandran and J. Wolfson's article Fill radius and fundamental group, whose main result is: Theorem. Let $N$ be a closed Riemannian manifold. If its universal cover has fill ...
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1answer
84 views

Connection on canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on symplectic manifold

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic 4-manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} ...
0
votes
1answer
202 views

curvature and volume growth

Let $M$ be a non-compact connected Riemannian manifold with $\mathrm{sec}_g=0$ and $\operatorname{vol} B(x,r)\geq c(n)r^n$ for any $r$, where $c(n)>0$. How to prove that $(M,g)$ is isometric to ...
4
votes
2answers
351 views

Infinite dimensional Riemannian geometry

My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...
2
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1answer
154 views

What is the meaning of Yang-Mills action evaluated on Levi-Civita connection?

On a Riemannian manifold $M$ with riemann curvature tensor $R_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R_{\mu\nu}dx^\mu\wedge ...
5
votes
1answer
285 views

A question on generalized Einstein metrics on four-dimensional manifolds

I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds, \begin{equation*} ...
2
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1answer
173 views

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^+ = ...
4
votes
2answers
333 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 ...
4
votes
0answers
311 views

Limit cycles as closed geodesics(geodesiable 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 ...
1
vote
2answers
320 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 ...
5
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0answers
67 views

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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0answers
74 views

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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3answers
223 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 ...
0
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1answer
71 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 ...
6
votes
2answers
289 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
votes
1answer
90 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 ...
3
votes
0answers
62 views

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 ...
28
votes
0answers
614 views

“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
votes
1answer
124 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 ...
1
vote
1answer
185 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 $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
0
votes
1answer
99 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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0answers
172 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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0answers
149 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 ...
3
votes
2answers
434 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 ...
0
votes
2answers
99 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
104 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 ...
7
votes
1answer
189 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 ...
1
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0answers
45 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 ...
3
votes
2answers
295 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 ...
12
votes
4answers
738 views

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 ...
3
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1answer
224 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 $V'$ ...
5
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1answer
527 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, ...
2
votes
1answer
98 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 ...
4
votes
2answers
179 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? ...
2
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0answers
66 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 ...
6
votes
2answers
378 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 ...
1
vote
1answer
114 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 ...
8
votes
3answers
357 views

Number of disjoint simple closed geodesics

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

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 ...
0
votes
1answer
204 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
votes
1answer
283 views

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 ...
7
votes
4answers
524 views

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 ...
2
votes
1answer
138 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 ...
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0answers
42 views

Rational homogenous functions

I'm interested in the set $\mathcal{S}$ of rational functions $F \colon \mathbb{R}^3 \to \mathbb{R}$ verifying: \begin{align} \Delta F=0 \quad \text{et} \quad F(\lambda x)= \lambda^d F(x) \quad d \in ...
6
votes
3answers
186 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 ...