A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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3
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2answers
269 views

Ricci Soliton geometric meaning

I wonder what is the geometrical, intuitive meaning of a Ricci soliton on a manifold. The definition that I use is as follows. $V$ is a vector field on the manifold, $g$ is a Riemannian metric. ...
2
votes
1answer
192 views

Reference request-What is the prerequisite of S.S.Chern's proof of the generalised Gauss-Bonnet theorem?

The title basically explains everything. The OP is an independent learner, who in the current stage sets S.S.Chern's proof of the generalised Gauss-Bonnet theorem as the goal. But what is the ...
2
votes
1answer
127 views

Intuition/visualization for a non-flat connection

I'd just like to check whether my visualization for a way to get a non-flat connection is correct. The definition I am using for a connection is, for a fiber bundle $\rho:E \to B$, a smooth ...
1
vote
1answer
45 views

For Green's function of $\Delta-c$, how to show $\int_{X}G(x,y)(\Delta-c)f(y) dy=\pm f(x)$?

Let $X$ be a compact Riemnnian manifold and $\Delta$ the Laplacian. Suppose that $G(x,y)$ be the Green's function of the elliptic operator $\Delta-c$ for a positive constant $c$. I think the ...
7
votes
0answers
196 views

Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$, i.e., each ...
2
votes
1answer
92 views

If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$

I'm having the same problem as it was questioned here. I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n ...
1
vote
1answer
48 views

Question on Normal Coordinates

I'm having a hard time trying to understand something that I'm suspicious is pretty stupid. I'll refer to Wikipedia to settle the term's I'll refer to. ...
1
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1answer
89 views

Multi Index Dirac delta function

If we assume the following result: $$\delta^{\alpha_1,\alpha_2,\cdots , \alpha_k, \rho}_{\beta_1,\beta_2,\cdots , \beta_k, \rho} = (n-k)\delta^{\alpha_1,\alpha_2,\cdots , ...
0
votes
1answer
197 views

Why the geodesic curvature is invariant under isometric transformations?

As I know the geodesic curvature $$ \kappa_g = \sqrt{det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ ...
5
votes
1answer
148 views

Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
1
vote
1answer
127 views

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
1
vote
2answers
136 views

Isometries and inner product in Hyperbolic SPACE (H3)

Well, the title says what I need to know/understand, when I studied upper half plane I remember that isometries are Mobius transformations (If I am not wrong), now I have no clue about it. Thanks for ...
1
vote
0answers
15 views

Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense for $M$ a compact Riemannian manifold?

Let $M$ be a compact Riemannian manifold. Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense?
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votes
1answer
20 views

Density of bounded functions in $L^1(0,T;L^1(M))$?

Let $u \in L^1(0,T;L^1(M))$ where $M$ is a compact Riemannian manifold. Is it possible to find $u_n$ such that $u_n \to u$ in $L^1(0,T;L^1(M))$ and $u_n$ are bounded everywhere or almost everywhere on ...
11
votes
2answers
634 views

Geometric meaning of symmetric connection

If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following: 1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0 2) ...
5
votes
2answers
245 views

Equivalent definitions of vector field

There are two definitions of a vector field on a smooth manifold $M$. A smooth map $V:M \rightarrow TM, \forall p \in M:V(p) \in T_p M$. A linear map $V:C^{\infty}(M) \rightarrow C^{\infty}(M), ...
0
votes
1answer
310 views

Problem in proving the property of Lie bracket of vector fields

Let $M$ be a Riemannian manifold, $f \in C^{\infty}(M)$, $X,Y$ vector fields on $M$. Then i have to prove $[X,f\cdot Y]=f\cdot [X,Y]+X(f)\cdot Y$. First i use the definition of Lie bracket: $[X,f\cdot ...
4
votes
1answer
87 views

The Levi-Civita connection in infinite dimensions

Is there an analogue of the Fundamental Theorem of Riemannian Geometry for (some subclass of) infinite-dimensional manifolds?
4
votes
1answer
69 views

“Bundle of metrics” on a principal bundle?

I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a ...
4
votes
1answer
90 views

Gauss curvature of C^2 surfaces

In do Carmo's book on Differential Geometry of Curves and Surfaces, the proof of theorema egregium, that the Gauss curvature of a surface immersed in $\mathbb{R}^3$ is invariant under local ...
0
votes
2answers
56 views

geodesic based on fixed points

Question: For any points $p,q\in M$, does there exist a geodesic curve connecting $p$ and $q$? Let $M$ be some constant curvature space, like $\mathbb R^n$, $\mathbb S^n$, $\mathbb H^n$. The answer ...
2
votes
1answer
23 views

Riemann integral and homoemorphism

I am wondering what happens if I have the following setup: I have a homeomorphism: $\phi$ from the unit sphere to the unit cube. I know that the characteristic function of the unit sphere is Riemann ...
2
votes
1answer
181 views

Totally geodesic and autoparallel

Let $M$ be a Riemannian manifold. A submanifold $N$ of $M$ is totally geodesic if every geodesic in $N$ is also a geodesic in $M$. On the other hand, $N$ is an autoparallel submanifold of $M$ if ...
0
votes
1answer
68 views

When can you recover a connection from totally geodesic submanifolds?

Let $g_{ab}$ a Riemaniann ( Lorentzian ) metric in a $n-$dimensional manifold $N$ and let $M$ be a submanifold of $N$. In general, the Levi-Civitta connection induced by the induced metric in $M$ ...
2
votes
1answer
114 views

Reference - Riemannian Orbifolds

I am looking for papers or textbooks talking about the various analog theorems of Riemannian Geometry of Manifolds to Riemannian Orbifolds like Toponogovs Theorem, Bonnet-Myers, Gauss Bonnet etc. So ...
1
vote
2answers
142 views

convexity and center of mass in riemannian manifolds

every riemannian manifold M is locally convex. Let $U$ be an open convex subset of $M$. Let $x_0, \ldots, x_n$ be points in $U$. Consider the map $\sigma \: \Delta \to U$ (where $\Delta$ is the ...
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vote
0answers
35 views

Weakest curvature assumption for existence of harmonic coordinates

Let (M, g) be a Riemannian manifold. What are the weakest curvature bounds for which one can construct harmonic coordinates on M (or at balls contained in M)? Does anybody maybe know if it is possible ...
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0answers
47 views

How to change the parametric equations of a hypersurface in $V_N$ to another form…

This exercise was given in the first pages of Synge & Schild Tensor Calculus. The parametric equations of a hypersurface in $V_N$ are $x^1=a\cos{u}$, $x^2 = a\sin{u^1}\cos{u^2}$, $x^3 = ...
2
votes
0answers
84 views

Seifert surfaces in Riemannian manifolds?

Does there exist an equivalent to Seifert surfaces for other Riemannian manifolds than $\mathbb{R}^3$? More precisely: Let $M$ be a simply-connected Riemannian manifolds and $K \subset M$ a (tame) ...
1
vote
1answer
178 views

Does a simply connected complete riemannian manifold with POSITIVE upper curvature bound have positive injectivity radius?

For example: I am thinking that some sort of rauch comparison theorem could be helpful
0
votes
1answer
44 views

Time evolution of Laplacian

While reading monograph on the Ricci flow, I came accross a fact (at least I think it is a fact), which is not proved explicitly in that book. Assume a smooth 1-parameter family of Riemannian metrics ...
1
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0answers
25 views

The time evolution of levi-Civita connection

Assume a smooth one-parameter family of Riemannian metrics $g_{t}$. Write $h:=\frac {\partial}{\partial t}g$. In addition, assume that the Levi-civita connection on the Riemannian manifold ...
4
votes
1answer
279 views

Question about Hopf-Rinow theorem

I'm studying Hopf-Rinow theorem and I don't see a step in the proof. Could someone help me, please? (Definition) Let's $(M, \langle,\rangle)$ an ANII(axiom numerability 2) and Hausdorff Riemannian ...
9
votes
3answers
955 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
0
votes
1answer
156 views

Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
1
vote
0answers
127 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
2
votes
2answers
111 views

Are all critical points of energy geodesics?

Let $\gamma$ be a smooth curve in a Riemannian manifold and consider the arclength functional $L(\gamma) = \int_a^b |\gamma'(t)|\, dt$ and the energy functional $E(\gamma) = \frac{1}{2}\int_a^b ...
2
votes
0answers
58 views

Why Riemannian metrics defining the same angles are conformal [duplicate]

Suppose $g_1$ and $g_2$ are two metrics defining the same angles, which means $g_1(X,Y)/(g_1(X,X)g_1(Y,Y))^{0.5}=g_2(X,Y)/(g_2(X,X)g_2(Y,Y))^{0.5}$ for all pairs of vector $X,Y$.I want to prove that ...
3
votes
0answers
230 views

Completeness of a Riemannian manifold with boundary

I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined ...
1
vote
2answers
246 views

Request for online reference to Hamilton's “The Ricci Flow on Surfaces”

Does anyone know of an online source for Richard Hamilton's paper "The Ricci Flow on Surfaces?" I've searched Google for it and it doesn't seem to give any results.
6
votes
1answer
240 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
4
votes
1answer
271 views

Klein Bottle Embedding on $\mathbb{R}^4$.

First of all, I am aware of the question in How to embed Klein Bottle into $R^4$ , which was inconclusive. Anyway, I've made some progress, but I still have a question. I am using Do Carmo's ...
1
vote
1answer
23 views

The higher-order estimates for the distance function

Let $M$ be a complete Riemannian manifold such that inj$(M)\geq l>0$ and $|\nabla^k\text{Rm}|\leq A_k$ for any $k \geq 0$ . For a point $p$ on $M$, we have a distance function $r(x)=d(x,p)$. For ...
1
vote
0answers
45 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
4
votes
1answer
169 views

Schwarzschild metric tensor normal vectors

The Euclidean Schwarzschild metric describing a manifold (a black hole, though this is not relevant to the question) is given by, $$\mathrm{d}s^2 = \left( 1-\frac{2GM}{r}\right)\mathrm{d}\tau^2 + ...
3
votes
1answer
71 views

The Lie derivative by an infinitesimal action is invertible at an isolated zero point

Let a compact Lie group $G$ act on a manifold $M$. Fix $X \in \mathfrak g$ and we write $X_M$ for the infinitesimal action of $X$. Assume that $p \in M$ is a zero point of $X_M$. Define a linear map ...
3
votes
1answer
67 views

Chain rule quesition: proving that the Weingarten map is self-adjoint

I'm reading through the proof in this paper (http://www.math.leidenuniv.nl/scripties/JaibiBach.pdf) but I'm stuck at the line: "Using the chain rule we get: $L_p(\phi_v) = -Dn(\phi_v) = - \frac ...
3
votes
1answer
354 views

Tensor Laplacian

For a general tensor $T_{\mu_1 \dots \mu_n}$ on a (pseudo-)Riemannian manifold, is it true that $\Delta (T_{\mu_1 \dots \mu_n})= (\Delta T)_{\mu_1 \dots \mu_n}$? In general, it is not true that ...
5
votes
1answer
162 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
2
votes
1answer
84 views

A lemma is John Lee's Riemannian Manifold having problem with proving it

The tangential connection on an embedded submanifold $M ⊂ R^n$ is symmetric. the hint is let $X,Y$ be vector fields that are tangent of M at points of M, so is $[X,Y]$ I start with $$T(X,Y)=\nabla_x ...