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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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 ...
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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 ...
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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: ...
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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.
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235 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 ...
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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 ...
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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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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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171 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 ...
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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 ...
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271 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 ...
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382 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 ...
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139 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 ...
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252 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 ...
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207 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 ...
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299 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 ...
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148 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 ...
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501 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 ...
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150 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 ...
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306 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
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1answer
260 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 ...
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95 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 ...
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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 ...
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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 ...
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171 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 ...
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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 ...
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229 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 ...
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166 views

Dose closed Alexandrov space admit a bi-Lipschitz embedding into $\mathbb R^N$?

as the title says. Let $A^n$ be an $n$-dimensional closed Alexandrov space. Does it admit a bi-Lipschitz embedding into the Euclidiean space $\mathbb R^N$ for sufficiently large $N$? I know there are ...
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654 views

About Sectional Curvature [closed]

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
3
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1answer
100 views

A k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of ...
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A question from Hamilton's Ricci Flow book by bennett chow

On page 3 of the book before exercise 1.2, is written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results ...
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geometric meaning of Ricci-flatness

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
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276 views

Strongly parabolic PDE vs weakly parabolic PDE

In my studies on the Ricci flow, I was faced with a problem. To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one ...
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2answers
557 views

Resolvent of Laplacian

Hello! Let $(M,g)$ be a Riemannian manifold and $-\Delta$ the Laplacian on M (acting on smooth functions). Then the resolvent $R(\xi)$ of $-\Delta$ is a compact operator. Is it possible to find for ...
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Bi-invariant Riemannian metrics on So(n)

Defining the inner product - 1/2 tr(XY) on the lie algebra so(n)(skew symmetric matrices) is one way to introduce a bi-invariant metric on So(n) since the inner product is ad-invariant. Are there any ...
2
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1answer
207 views

Different notions of geodesics.

Let $M$ be a (without boundary and not necessarly complete) Riemannian manifold. A map $c\colon [a,b]\rightarrow M$ is called geodesic of type A iff $c$ is piecewise smooth, parametrized proportional ...
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862 views

Difference between parallel transport and derivative of the exponential map

This is a crosspost from math.stackexchange Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then ...
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Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari. In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
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400 views

Is it true that the geodesics on SO(n) and SU(n) are closed?

I mean for the bi-invariant metric (but actually any metric would work). In this metric geodesics are translates of 1-parameter subgroups so we need only to show that $exp(t X)$ for any X in the lie ...
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286 views

Normal coordinates

On a Riemannian Manifold $(M,g)$, for any $p \in M$, there exists a normal coordinate system at p, ${U, x_1, ..., x_n}$. I want to integrate over a normal coordinate system with U being a geodesic ...
2
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1answer
208 views

positive sectional curvature of submanifold in $R^n$?

Let $N$ be a hypersurface in $\mathbb R^n$, assume it is compact. Then the maximum point of $d(O, x)$ when restrict to $N$ has positive sectional curvature lower bound by the one of the correspond ...
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Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious. General gist of the problem I have a variational problem on a ...
2
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1answer
199 views

Diameter estimate of distance sphere of positive curved manifold

Let $M$ be an $n$-dimensional Riemannian manifold with sectional curvature lower bound 1. Fix a point say $O\in M$, let $S(r)$ denote the distance sphere centered at $O$ with radius $r$. The classical ...
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Special coordinates for periodic metrics

This question is a follow-up to that one. Given a $\mathbb{Z}^n$-periodic metric $g$ on $\mathbb{R}^n$ (with $n>2$), is it possible to find a periodic diffeomorphism $\varphi$ such that ...
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Reference - Asymptotic geodesics on compact surfaces without conjugate points

I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of ...
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Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood

Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
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The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$

Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...
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Induced Riemannian metric on Jet-Manifold

Suppose $(M,g)$ and $(N,g')$ are smooth Riemannian manifolds and $J^r(M,N)$ is the smooth manifold of $r$-jets $j^r_xf$ of smooth maps $f:M\to N$. Is there an 'induced' Riemannian metric $g''$ on ...
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$C^k$ topology of metrics

Is the space of Riemannian metrics, over a compact manifold, complete when endowed with the $C^k$-topology of metrics?. Is there a good reference for this?
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Hessian of the inverse exponential map on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold. Then define $f: T^*M \times M \to \mathbb{R}$ $f(x,\xi, y) = \langle exp_x^{-1} y, \xi \rangle$ where $exp_{\cdot}\cdot$ is the the exponential map and it's ...