Tagged Questions

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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1answer
101 views

Integral estimate on a two dimensional Riemannian manifold

For my Master's thesis, I'd like to prove the following (but I'm not sure it's true): On a two-dimensional Riemannian manifold (oriented and closed), for any smooth function $f$, it holds that $$ ...
4
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0answers
251 views

Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements $\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...
6
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1answer
609 views

How metric is Riemannian geometry

Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by $$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, ...
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1answer
492 views

Canonic identification of the tangent space of the Grassmannian

let $Gr(k,V)$ be the grassmannian of k-dimensional subspaces of the complex vector space $V$ of dimension $n>k$. I know that, given $K\in Gr(k,V)$, $T_{Gr(k,V),K}\simeq Hom(K,V/K)$, but i want to ...
5
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1answer
252 views

Riemannian and symplectic structures

Let $(\mathcal M,g)$ be a smooth Riemannian manifold and $\Delta$ be the standard (positive) Laplace operator given in coordinates by the usual $$ \Delta=-\vert g\vert^{-1/2}\partial_j(\vert ...
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1answer
201 views

choices of connection in prequantization

In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...
3
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2answers
486 views

Van Vleck-Morette Determinant

There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source ...
3
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2answers
221 views

Ito Diffusions with low regularity?

I would like to have an Itô Diffusion $$ X_t = \int_0^t b(s) \mathrm{d}s + \int_0^t \sigma(s) \mathrm{d}B_s.$$ where the (vector- and matrix-valued, respectively) functions $b$ and $\sigma$ have lower ...
3
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1answer
823 views

Proof of a theorem of Jean-Pierre Serre on geodesics of closed Riemannian manifolds

An oft-cited theorem of Serre states that there are infinitely many geodesics between any two points in a closed Riemannian manifold. Could someone please provide an intuitive sketch of the proof?
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1answer
138 views

pre-symplectic and foliation and its trajectories

Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?
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1answer
158 views

A question on asymptotically flat metrics

For $M$ a Riemannian manifold, with Riemannian metric $g$ and $x$ a point in M, what is the meaning of "$g$ on $M\backslash\{x\}$ has an 'asymptotically flat end at $x$'."? (See this paper on page 16, ...
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2answers
449 views

Equality of the determinants of certain submatrices of an orthogonal matrix

Is the determinant of any submatrix of an ORTHOGONAL matrix extracted from the intersection of $k$ row and $k$ columns equal to that of the $(n-k)(n-k)$ submatrix remaining after deletion of these ...
7
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1answer
372 views

Quantum Cellular Automata on Riemannian manifolds and geometric group theory

We try to motivate our question. We have a certain logical/operational structure that has an emergent physical interpretation. We are giving this structure a geometric setting via quasi-isometries. ...
3
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2answers
370 views

Aubin's book - construction of Green's function on compact manifold

In Aubin's book (nonlinear problems in Riemannian Geometry), starting from p. 106, it is shown that a Green's function of a compact manifold without boundary satisfies $$G(P,Q) \leq k ...
11
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1answer
646 views

Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?

Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity. Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ ...
6
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0answers
219 views

Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside? Basically, I am asking for a ...
2
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1answer
424 views

Existence of Geodesics in continuous metrics

I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the ...
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0answers
117 views

Nash embedding with target which is not $\mathbb{R}^{m}$

I'm curious about the following question: Given $(M^n,g)$ a closed Riemannian manifold, is there always a $C^\infty$ isometric embedding $F:(M^n,g) \to (\mathbb{S}^{m},g_{std})$ for $m$ large ...
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2answers
352 views

Computations with the distance function on a Riemannian manifold

Let $(M,g)$ be a complete Riemannian, connected, compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then $$ ...
3
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2answers
159 views

Random metrics on compact orientable surfaces

Hello everyone, Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of ...
4
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0answers
143 views

Gromov-Haussdorf and Lipschitz convergence of a non-collapsing sequence of manifolds with Ricci curvature bounded below

There is a theorem from Cheeger-Colding saying the following: Let $n$ be an integer. If a sequence of $n$-dimensional Riemannian manifolds $(M_i,g_i)$ converges with respect to the Gromov-Hausdorff ...
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2answers
976 views

Torsion and Parallel Transport

There's a close relationship between curvature and the holonomy group; the holonomy theorem of Ambrose and Singer, for example. It seems to me that there should be an analogous result for torsion. I ...
5
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2answers
340 views

Negative sectionnal curvature and constant curvature

Good morning everyone, I was wondering about the difference between manifolds carying a Riemanniann metric with negative sectionnal curvature and hyperbolic manifolds. I was told once "there are ...
0
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1answer
224 views

Why don't $\mathbb{T}^n, \mathbb{S}^n, \mathbb{H}^n$ admit other metrics of constant curvature?

The torus $\mathbb{T}^n$, the sphere $\mathbb{S}^n$ and the hyperbolic space $\mathbb{H}^n$ admit metrics of constant (sectional) curvature $0, 1, -1$ respectively. Do they afford metrics of constant ...
6
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1answer
460 views

Geometry defined by foliation.

In $\mathbb R^3$ there are 3 natural foliations given by the lines parallel to each axis, which intersect transversally. Let $M^n$ a manifold with $n$ foliations by lines or circles that intersect ...
7
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1answer
459 views

complete metric space

Hallo, I have the following question: Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: ...
3
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1answer
234 views

In the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms

Can anyone help me and prove that in the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms? Thanks for your time.
2
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1answer
237 views

Time has dimension $2$ with respect to the Ricci flow scaling

Terence Tao in his lecture notes on Ricci flow has written: If we are to find a scale-invariant (and diffeomorphism-invariant) monotone quantity for Ricci flow, it had better be constant on the ...
3
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0answers
532 views

On Perelman's paper

In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written: Fix a closed manifold $M$ with a probability measure $m$, and suppose that our ...
3
votes
1answer
143 views

Volume growth of covers and growth of deck-transformation groups

It is well-known that if $\widetilde M\to M$ is a Galois cover of a compact Riemannian manifold $M$ with deck-transformation group $G$, then the growth of $G$ equals the volume growth of $\widetilde ...
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2answers
150 views

Reference request: Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$

Does anyone know a citeable reference which works out the properties (geodesics, geodesic distance, ect) of the Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, ...
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2answers
172 views

Can one (block) diagonalize the curvature matrix of 2 forms on a Riemannian manifold?

Let $M$ be a smooth Riemannian manifold, let $R$ be the Riemannian curvature operator, and let $p$ be a point in the manifold. With respect to any orthonormal basis of the tangent bundle at the point ...
5
votes
1answer
231 views

Invariants of a $GL(3,\mathbb{R})$ action

I'm trying to understand the standard $GL(3,\mathbb{R})$ action on the 15-dimensional space of possible values for the derivative of the Riemann curvature tensor of a 3-dimensional manifold $M$ at a ...
3
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2answers
272 views

Positively curved manifold with a codimension 1 totally geodesic submanifold.

Fact : Consider the inclusion $V^{n-1} \rightarrow M^n$ where $M$ is a closed orientable simply connected positively curved manifold. Then connectivity lemma implies that the inclusion is ...
1
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1answer
388 views

Ricci flow as a gradient flow and its Lyapunov function

In study of Ricci flow, for making Ricci flow as a gradient flow I faced $\mathcal{F}(g,f)=\int (R+|\nabla f|^2)e^{-f}$. I know that if we suppose $\frac{df}{dt}=-R$, then ...
1
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1answer
140 views

Holonomy groups of quotient Riemannian manifolds?

Let $(X,g)$ be a Riemannian manifold with holonomy group $Hol(X,g)$. Suppose that a finite group $G$ acts on $X$ freely and the metric $g$ is invariant under $G$. What can one say about the the ...
4
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1answer
258 views

Alexandrov angles in Riemannian manifolds

Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch ...
9
votes
1answer
210 views

Positively curved manifold with almost extreme diameter

Suppose $M$ is a 1-connected closed manifold with sectional curvature $\ge 1$. So the diameter $D$ of $M$ satisfies $$ D \le \pi $$ When equality holds $M$ is isometric to round sphere. In fact this ...
4
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2answers
309 views

Riemannian manifolds with small geodesics and bounded curvature

Let $(M,g)$ be a compact riemannian manifold with sectional curvature $|K_g| \leq 1$. A lemma due to Klingenberg asserts that then either the injectivity radius $i_g \geq \pi$ or $(M,g)$ contains a ...
3
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1answer
151 views

Is geodesic plane field a Killing field?

Let $M$ be a closed orientable Riemannian manifold. Recall that a plane field on a Riemannian manifold is said to be geodesic if any geodesic tangent to the plane field at one point is tangent to it ...
3
votes
2answers
510 views

Energy functional

During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...
1
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1answer
154 views

Homogeneous Spaces and Equivariant Hodge Maps

For a homogeneous space $G/H$, endowed with a $H$-equivariant metric $g$, let $\ast$ be the corresponding Hodge star map. It seems that $\ast$ must also be $\ast$-equivariant, but I can't see how one ...
3
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1answer
316 views

On Dimension of Instanton Moduli Space

I am reading Charles Nash's book on differential topology and QFT. In particular, I have question on the part calculating dimension of instanton moduli space. The question split into conceptual part ...
4
votes
1answer
269 views

Prescribing the Lie derivative of the metric?

This is a question that arises from my research problem. Suppose $(M,g)$ is a compact Riemannian manifold with boundary and $g$ is smooth up to the boundary (if you like, take $M$ to be diffeomorphic ...
6
votes
1answer
97 views

Heat Kernel Asymptotics with low regularity

Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous. Question: Is there still an asymptotic expansion of the heat kernel of the form $$ p_t(x, y) \sim (4 ...
8
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4answers
1k views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
1
vote
1answer
75 views

3-dim 1-connected Alexandrov manifold with curvature $\ge 0$ Heomomorphic to sphere?

For Alexandrov manifold in the title we mean 3-dim Alexandrov apace which is also a topological. manifold. Shioya-Yamaguchi posted a conjecture on their paper "Collapsing 3-manifold with lower ...
3
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1answer
172 views

Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold

Hallo, I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ...
2
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2answers
162 views

Geometry of Hopf fibrations and the fibration of Steifel Manfiolds over Grassmannians

When $F = \mathbb{R}, \mathbb{C}$ or $\mathbb{H}$, there are fibrations $$O(k,F)\rightarrow V_k(F^n)\rightarrow G_k(F^n)$$ where $V_k(F^n)$ are Steifel manifolds and $G_k(F^n)$ are Grassmannians. When ...
2
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1answer
233 views

Is there a lower bound for variance in terms of curvature?

If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero ...