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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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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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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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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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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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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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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25 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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28 views

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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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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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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41 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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24 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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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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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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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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48 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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40 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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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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45 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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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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45 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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30 views

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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51 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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39 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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59 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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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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28 views

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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42 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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40 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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42 views

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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82 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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36 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 ...
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Maximal geodesics on compact manifolds

I have two questions about the following passage in Taubes's book on differential geometry. I also quote the proposition it references. 9.1 The maximal extension of a geodesic Let $I \subset R$ ...
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Master's Exploration in General Relativity

just throwing a query out to the Math community. I'm about to embark on a master's in Gravitation, Cosmology and General Relativity and was looking for possible subjects to start researching. My main ...
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Showing a vector field is smooth.

Let $(M,g)$ be a Riemannian manifold, $N$ a smooth manifold and $$\pi:M\to N$$ a surjective smooth submersion. Then, each level set $M_q=\pi^{-1}(q)$ is a properly embedded submanifold of $M$ so we ...
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Why does it suffice to check the geodesic equation to leading order?

I am reading Taubes's book on differential geometry and am wondering about a proof. My apologies if this is simple, as I'm still grappling with the material. My question concerns material in chapter ...
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Tubular neighborhood by restricting the Riemannian exponential map

Let $M$ be a Riemannian manifold (possibly non-compact, possibly non-complete) and $N\subseteq M$ a smooth submanifold (possibly non-compact). Does there exist a continuous $\mu\colon M\rightarrow ...
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Show that a nonconstant subharmonic function on a manifold cannot attain its supremum

PROBLEM: Suppose $f$ is a smooth non-constant function on a connected Riemann manifold $M$ of dimension 2 such that $f$ is bounded and $\Delta_M f \ge0$. Show $f$ cannot attain its supremum. I try ...