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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Space of $G$-connections; respecting a spin structure

If I want to have a space of $G$-connections on a Riemann surface, I can take the fundamental group on the surface, represent its generators on $G$ and take (up to conjugation) those representation ...
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Computing curvature of hyperbolic space

Consider the unit ball in $\mathbb R^2$ endowed with the Poincaré metric: $$ds^2=\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}.$$ I want to compute the Gaussian curvature and find that it is $-1$. Given that ...
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Chern-Gauss-Bonnet theorem for even-dimensional manifolds with boundary

On the wikipedia page for the Chern-Gauss-Bonnet theorem it states that there is a generalization of the theorem for even-dimensional manifolds with boundary, but does not provide the relevant theorem ...
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Constructing a metric for the tautological line bundle of $\mathbb C P^2$

I'm doing some independent reading in differential geometry, and the following is my attempt to work out the details of the construction of the tautological bundle on $\mathbb CP^2$ and the induced ...
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1answer
57 views

Is Hilbert's theorem generalizable to $H^3$ immersion in $\mathbb{R}^4$?

The Hilbert theorem in differential geometry concerns the immersion of the hyperbolic plane in $\mathbb{R}^3$. Is it valid for $H^3$ in $\mathbb{R}^4$?, and for all $H^n$ in $\mathbb{R}^{n+1}$?
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Complete Riemannian metric on ${\mathbb R}^2\setminus\{0\}$.

It seems to me that the Riemannian metric $g_{ij}=\delta_{ij}/|x|^2$ on the punctured plane is complete, but I don't find a proof not involving explicit computations of the geodesic equation. Does ...
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Taylor series representation for a Riemannian hypersurface

This is Exercise 8.5 in Lee's Riemannian Manifolds: An Introduction to Curvature. Suppose $M\subset \mathbb R^{n+1}$ is a hypersurface with the induced metric. Let $p\in M$, and let ...
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a problem in gauss lemma

I was reading the Gauss Lemma from the do carmos Rienmannian geometry book which says that Let $p \in M$ and let $v \in T_pM$ such that $\exp _p v$ is defined Let $w\in T_pM$ is identified with ...
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28 views

Equidistant points to a hyperbolic line

consider the Poincare upper half-plane model of hyperbolic plane $\mathbb{H}^2$ and a hyperbolic line $\ell\subset \mathbb{H}^2$ (or geodesic if you want). I would like to visualize the set of points ...
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looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
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11 views

Geodesic connectivity implies geodesic convexity?

Assume a subset $C$ of a Riemannian manifold $M$ is "geodesically-connected", that is: given any two points in $C$, there is a geodesic contained within $C$ that joins those two points. Is it true ...
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81 views

Distance of two hyperbolic lines

Consider the upper-half plane model of the hyperbolic plane $\mathbb {H}^2.$ Now consider two lines in it given as $\ell_1:=\lbrace { (x, y)\in \mathbb {H}^2 \vert x^2 +y^2=r^2\rbrace}, ...
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1answer
19 views

Countable intersection of Cut Locuses is always empty?

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then does there always exist a countable collection of points $\{p_n\}_{m \in \mathbb{N}}$ such that: \begin{equation} ...
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If $N$ is the boundary of Riemannian $M$, can I compute $i^{*}(* (\alpha \wedge \beta))$?

There wasn't enough room in the title to explain completely: $M$ is an oriented Riemannian manifold with boundary $N$. $\alpha$ and $\beta$ are differential forms on $M$, $*$ denotes the Hodge star, ...
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41 views

Classical differential operators with complex functions on Riemannian manifolds

I am having some trouble understanding how to use the classical operators ($\nabla, \operatorname{div}, \Delta$) with complex functions on a Riemannian manifold $(M, g)$. Consider the formula ...
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29 views

2-dimensional Riemann Manifold

I am looking for a proof of the theorem that states that any 2-dimensional Riemann Manifold is conformally flat in the case of a metric of signature 0, following through with Problem 6.30 in the text ...
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3answers
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The Riemannian Distance function does not change if we use smooth paths?

The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all piecewise smooth paths between $p$ and $q$. Does it change if we take the infimum only over smooth ...
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36 views

Computation in Wikipedia's article “Riemann Curvature Tensor”

This Wikipedia article explains how the Riemann curvature tensor is a measure of the failure for a tangent vector to parallel translate back to itself along an infinitesimally small loop. The article ...
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40 views

Triangle equality in a Riemannian manifold implies “geodesic colinearity”?

Let $(M,g)$ be a non-complete Riemannian manifold. Assume $p,q,r\in M$ satsify: $d(p,q)=d(p,r)+d(r,q)$, where $d$ is the Riemann distance function induced by the metric $g$. I am trying to find the ...
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1answer
43 views

Diffeomorphism between $\Bbb{R}^{4}$ and the cube

I'm looking for an explicit diffeomorphism between the four-dimensional euclidean space $\Bbb{R}^{4}$ and the four-dimensional open cube. I wonder whether there is a simple looking map, with simple ...
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1answer
25 views

Intersection of Cut Locuses

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then when is: \begin{equation} \bigcap_{p\in M} C_p(M)=\emptyset\text{ ?} \end{equation}
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113 views

Riemannian metrics and how spaces look

I thought I had a fairly good understanding of Riemannian metrics until I came across this exercise in Petersen's book. Construct paper models of the Riemannian manifolds ($\mathbb{R}^2, dt^2 + ...
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geodesic of Stiefel manifold

Define a metric on Stiefel manifold $V_{n,p}$ as $$\left<\Delta_1,\Delta_2\right>=\text{tr}\Delta_1^T\left(I-\frac{1}{2}YY^T\right)\Delta_2$$ $\forall \Delta_1,\Delta_2\in T_YV_{n,p}$ how to ...
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36 views

N-polygons in hyperbolic geometry

Let $N$ be an integer and we have two $N$-polygon $A_{1}A_{2}\ldots A_{N}$ and $A'_{1}A'_{2}\ldots A'_{N}$ such that the length of geodesic $A_{i}A_{i+1}$ is equal to the length of geodesic ...
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20 views

Reference needed for a short time existence result of quasilinear PDE on a compact manifold (relating to Ricci flow).

I'm currently in the proces of learning and writing a bit about the Ricci flow. In particular I'm studying the case of compact 2d Riemannian manifolds. Mostly I'm making good progress but I do miss ...
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2answers
53 views

Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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1answer
41 views

Which Riemannian Manifolds are scale-free?

Let us call a Riemannian manifold $(M,g)$ scale-free if for any real positive scalar $\lambda$, $(M,g),(M,\lambda g)$ are isometric. $\mathbb{R}^n$ with the standard metric $g$ is scale-free. (Via ...
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103 views

Lie bracket is part of the intrinsic “geometry”? But I have seen it defined without a metric…?

I have seen two definitions of the Lie Bracket for a Riemannian manifold $(M,g)$. One is this : $[X,Y] = D_X Y - D_Y X$, where $D$ stands for covariant differentiation. When written out, this seems ...
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1answer
48 views

Curve with with curvature $k(s)\ge 1$ everywhere has diameter $\le 2$

Let $\alpha(s)$ be a simple closed plane curve. Define the diameter $d_\alpha$ of $\alpha(s)$ to be $$d_\alpha = \sup_{t,s\in\mathbb R} \| \alpha(s) - \alpha(t) \|.$$ Assume the curvature ...
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49 views

Local isometries preserve geodesics?

Question: It is well known that if $\varphi:M\to \tilde{M}$ is an isometry between Riemannian manifolds, then $\varphi$ maps geodesics of $M$ to geodesics of $\tilde{M}$. I am wondering if it is ...
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Question of well-definedness of the Levi-Civita connection?

On page $55$ of Do Carmo's Riemannian geometry, he proves that there is a unique symmetric affine connection compatible with a given metric on a manifold M. He defines it by a formula $\langle Z, ...
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Riemannian Geometry notational tricks or alternatives

I am interested in learning tricks that people have developed to speed up / clean up calculations in Riemannian Geometry. I am hopeful about this question because there is often a lot of symmetry in ...
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2answers
49 views

Integration over Riemannian Manifolds

Can we integrate over non-orientable riemannian manifold? If so, how do we do it? Some references would be nice. Thank you!
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1answer
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How to express curvature of a level set in terms of derivatives of a function?

Suppose I have a smooth function $u:\mathbb R^n\to\mathbb R$. Assume that its gradient doesn't vanish (near any point where we investigate it). Is there a list of different (intrinsic and extrinsic) ...
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58 views

Orientability of Surfaces and the Fundamental Group

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...
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1answer
43 views

Hodge laplacian of distance function

Let $p$ be a given point on a Riemaniann manifold $\mathcal{M}$. The distance function to point $p$ is denoted $f_p$ : $$ f_p(q) = \operatorname{dist}(p,q)$$ The exterior derivative is denoted ...
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Smooth isometric embeddings of Riemannian manifolds

The essence of this question is: Let $(M,g_M)$ and $(N,g_N)$ be Riemannian manifolds. How many different ways are there to embed $M$ isometrically in $N$? In this context, I say the embedding $i_1$ ...
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Prove that an homogeneous and isotropic Riemannian manifold has constant sectional curvature

I have problems in proving that an homogeneous and isotropic Riemannian manifold has constant sectional curvature. This is my attempt: By definition, the manifold $M$ has constant sectional ...
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1answer
65 views

Describe the Riemann surface:

$$W = \sqrt{1-z^2}$$ I would like hints only. Using @Dr.MV's hint, I get two factors: the first is $$\sqrt{(x-1)+y^2}^{\frac{1}{2}}e^{i\frac{\theta}{2}}$$, which, when we let theta range from 0 to ...
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1answer
62 views

Isometry from warped product onto the base.

Let $B$ and $F$ be semi-Riemannian manifolds with metric tensors $g_B$ and $g_F$, and consider the warped product $B \times_f F$ by a smooth map $f: B \to \Bbb R$, with metric tensor: $$g = \pi^\ast ...
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1answer
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Proving that this metric tensor is Riemannian

Let $(M,g)$ be a Riemannian $n$-manifold, and $\varphi: M \to \Bbb R$ be a smooth map. Define another metric tensor by: $$\widetilde{g} = g - {\rm d}\varphi \otimes {\rm d}\varphi$$I know that ...
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1answer
43 views

Nice expression for curvature of a Kähler surface

Let $\Sigma$ be a Riemann surface with symplectic form $\omega$ and complex structure $J$, and denote by $g$ the induced metric. My question is Is there a nice expression of the Gaussian curvature ...
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1answer
41 views

Geodesic vector field is well-defined

Let $(M,g)$ be a Riemannian manifold. I just learnt that for a curve $x:I\to M$ to be a geodesic, the geodesic equation $$\ddot{x}^k+\dot{x}^i\dot{x}^j\Gamma^k_{ij}=0$$ is equivalent to the condition ...
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Difference between exponential maps composed with parallel transport along two different geodesics?

Let $(M,g)$ be a Riemannian manifold, and let $\gamma_{p,v}, \gamma_{p,w}$ be two geodesics starting from $p$ with directions/initial vectors $v,w$ respectively. Consider the two operations (to be ...
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Maximally symmetric manifold with boundary and non-vanishing extrinsic curvature?

I was wondering if the following requirements are compatible: Given a $d$-dimensional manifold with boundary $M$ with $\partial M\neq \emptyset$ endowed with a metric $g$. The following conditions ...
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1answer
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Two different definitions of a Liouville measure

Ok, I'm currently confused because of two different definitions for the Liouville measure associated to a smooth manifold $M$ of dimension $n$. These are: a) The measure $\mu$ on the cotangent bundle ...
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Show that $(divR) (X, Y,Z) = (\nabla_X Ric) (Y,Z) − (\nabla_Y Ric) (X,Z).$

In course of solving Riemannian Geometry By Peter Petersen Chap. 2, I stuck on the following problem: Show that in a Riemmanian manifold if $R$ is the $(1, 3)$ curvature tensor and $Ric$ the $(0, ...
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2answers
84 views

Curvature flow for convex planes curves

Tentative translation of the original question. I've read several articles on the curvature flow for convex plane curves (the curve remains convex during evolution, and eventually shrinks to a point). ...
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1answer
39 views

Heat kernel formula on hyperbolic plane well defined

Consider the heat kernel for the hyperbolic plane $\mathbb{H}^2$ and the corresponding heat kernel: $$k(x,y,t)=\frac{C}{t^{\frac{3}{2}}}\cdot ...
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Are bounded geodesics in the modular surface closed?

Let $M=\mathbb{H}/SL(2,\mathbb{Z})$ be the modular surface (which is noncompact but finite volume with the volume induced by the constant negative curvature metric inherited from $\mathbb{H}$). Any ...