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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Geodesics on Lorentzian (2n-1)-Spheres

I know that if we endow $S^{n}$ with the round Riemannian metric, we will be able to join the North pole and the South pole by an unlimited number of geodesics, in particular the meridians, and indeed ...
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Characterization of locally conformally flat manifolds with Frobenius theorem

In Hamilton's Ricci Flow (by Chow, Lu, Ni, pp. 29-31, see here) they show that a Riemannian manifold $(M^n,g)$ is locally conformally flat iff the Weyl tensor vanishes (when $n\ge 4$) and iff the ...
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Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
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Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
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Connectedness and Hopf-Rinow Theorem

Does the Hopf-Rinow theorem hold if the Riemannian manifold is not necessarily connected? $\\$ $\bf{Motivation \ and \ Minor \ Details \ About \ Question:}$ I am reading a non-standard book which ...
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How to prove that $\forall$ Riemannian manifold $M$, $T\in C^{1}(M)$ and $x,y\in M$ close enough: $d(Ty,exp_{Tx}(D_xTexp^{-1}_xy))\leq d(x,y)$?

How to prove that $\forall$ Riemannian manifold $M$, $T\in C^{1}(M)$ and $x,y\in M$ close enough: $$d(Ty,\exp_{Tx}(D_xT\ \exp^{-1}_xy))\leq d(x,y)$$ My attempts so far were only able to show the ...
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An application of Nash's embedding theorem to manifolds with fixed volume form

I have a smooth (possibly compact, or closed, or oriented, or more than one of the previous) $n$-manifold $M$ together with a fixed volume form $\rho\in\Omega^n(M)$. Can $M$ be embedded into some ...
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Riemann metric in the open disk

I am currently studying The Princeton companion to mathematics. According the book, "A more precise definition is that the open unit disk is the set of all points $(x,y)$ such that $ x^2 + y^2 < ...
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What is the differential of left translation?

Let $G$ be a Lie group, $g\in G$ and $L_g$ be left translation by $g$. I want to compute the differential $dL_g|_0$ of $L_g$ at $0$. Attempt: Let $v\in T_0G$ be a tangent vector at $0$. Let ...
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The normal curvature is bounded by the principal curvatures.

Let the inclusion $i:S\subset\mathbb R^3$ be an immersion of a surface $S$, and let $N:S\to \mathbb R^3$ be a local Gauss map. Let $a:I\to S$ be an arc length parametrized curve, with $a(0)=p$ and ...
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Are all 2D tensors in a specified flat metric equal to that same metric conformally scaled?

I have a tensor $T_{mn}$ where its indices coorespond to a flat metric $g_{mn}$. I want $T_{mn}$ to be a new metric $\tilde{g}_{mn}$, such that $T_{mn}(g_{rs}) = \tilde{g}_{mn}$. A theorem says that ...
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Reference: In every free homotopy class is a unique minimizing closed geodesic

Does anyone know a reference for the following result: Let $M$ be a compact hyperbolic manifold/manifold with strict negative curvature . Then in every non-trivial free homotopy class of $M$ there ...
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About prime geodesic cycles and deck transformations group

I'm proving theorem 2 occurring in Sunada's paper Riemannian coverings and isospectral manifolds. Unfortunately Sunada's quotes himself to the following paper: Tchbotarev’s density theorem for closed ...
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Petersen Riemannian geometry p86

I'm confused by a computation in Peter Petersen's Riemannian geometry book. We consider $S^{2n+1}$ viewed as embedded in $\mathbb{C}^{n+1}.$ The circle $S^1$ acts naturally on $S^{2n+1}$ by complex ...
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If $\dfrac{\mathrm {circumference}}{\mathrm {diameter}}$ is the same for all circles, does the surface have to be flat?

Given a two dimensional Riemannian manifold with the property that the ratio of the circumference and the diameter is the same for all circles. What can be said about it? Does it have to be the ...
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Complete Riemannian metrics in cylinder $\mathbb{R}\times X$ and cones $\mathbb{R}^{+}\times X$

Consider the cylinder $\mathbb{R}_t\times X$ where $X$ is a compact manifold without boundary. Consider the cylindrical metric $g_{cyl}=g_X+dt^2$. Clearly $(\mathbb{R}_t\times X, g_X+dt^2)$ is a ...
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Introductory Text about Riemannian Manifolds

Some of my friends and me want to study the subject of Riemannian manifolds, and we are looking for an introductory text to study that subject. We studied differential geometry, and are about to ...
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Upper bound on the distance of orthogonal matrices

Dear math stackexchange users, I have a question on orthogonal matrices: suppose I have a matrix $X\in\mathbb{R}^{n\times n}$ and I consider the orbit of the orthogonal group $O(n)$ acting from the ...
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21 views

Reference on Loop Space

I need to study the foundations of the theory of closed loop spaces. I have been referenced to Klingenberg's "Lectures on Closed Geodesics", but found it a dry and difficult reading. Is there some ...
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24 views

How does a Lie derivative generate a $U(1)$ isometry?

Consider a $2l$-dimensional Riemannian manifold $(M,g)$ without a boundary and let $V=V^{\mu}\frac{\partial}{\partial^{\mu}}$ be a Killing vector field, i.e. $$ \mathcal{L}_Vg_{\mu \nu} = 0 ...
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Why don't we write $\nabla_{X}(fY) = f\nabla_{X}Y$ instead of $\nabla_{X}(fY) = f\nabla_{X}Y+ X(f)Y$ for affine connections?

According to do Carmo, in Riemannian Geometry pages 49-50, he says let $\mathcal{X}(M)$ denote the set of all vector fields of class $C^{\infty}$ on $M$. Let $\mathcal{D}(M)$ denote the ring of all ...
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Riemannian Volume vs. Euclidean Volume

If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi \,d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; ...
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Stereographic projection is conformal in the sense of bilinear forms?

This is a past exam problem from my university. However, the corresponding course sequence does not cover any Riemannian geometry, so I'm not sure how to go about this so much. Let ...
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Problem with the concept of connection

I've been told that there is only a canonical way for doing the vertical subspace of the tangent bundle of a manifold and in order to do the horizontal subspace you need a connection. These are very ...
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Convolution of functions defined on manifold

Let $M$ be a Riemannian manifold with fixed volume form $\mu$. How to define a convolution of two 'functions' $f,g \in L^1(M)$? I will be grateful for an answer or for giving me some refrence where it ...
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Measures which are absolutely continuous with respect to a Riemannian measure

Suppose $(M,g)$ is a oriented connected Riemannian manifold (but not necessarily compact). Let $\omega_g$ denote the volume form on $M$ determined by $g$, and let $m_g$ denote the probability measure ...
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How to make sure any two points with small enough distance are inside a common open set

Let $K$ be a compact subset of a metric space, and I cover $K$ with finite open sets. How can I select $\delta>0$, such that for any $x,y\in K$, with $d(x,y) <\delta$, $x,y$ are inside an open ...
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47 views

Role of Group actions in Differential Geometry

This is a rather soft question, my hope is to bring some order into the stuff I would like to learn about differential geometry -- here it is: I was told over and over again that Geometry has to do ...
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Is there a charaterization of riemannian product manifolds?

Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...
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Riemannian Metric of Lobatchchevski Geometry

I am stuck at a problem from Riemannain Geometry, written by Do Carmo. A function $g:\mathbb R\to\mathbb R$ given by $g(t)=yt+x$, $t$,$x$,$y\in\mathbb R$, $y>0$, is called a proper affine ...
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Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by ...
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Global Bi-harmonic functions in a Riemannian manifold

Any help will be appreciated thanks! Consider $(\mathbb{R^n},g)$ to be a Riemannian manifold. For simplicity we can assume the manifold to be asymptotically euclidean outside a compact domain ...
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Deriving Ricci identity for co-vector fields

Let $\nabla$ be the covariant derivative associated with a torsionless connection. Prove the Ricci identity for covectors: $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = ...
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Why the equality of spectral zeta functions imply the isospectrality?

Let $\Delta_{M_1}$ and $\Delta_{M_2}$ be the Laplace-Beltrami operators on two compact and connected Riemannian manifolds $M_1$ and $M_2$ respectively. We define the spectral zeta function (or ...
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Why is the local representation of a connection a projection on $T_{\xi}E$?

In Klingenberg's Lectures on Closed Geodesics, he states a proposition that goes as follows: Proposition: A connection $K$ on $\pi: E \rightarrow M$ defines a splitting $TE=T_hE \oplus T_vE$ ...
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Naive question about Hermitian metrics

Let $M$ be a complex manifold with complex structure $J$ and Riemannian metric $g$. Then I know that $g$ is said "Hermitian" if it satisfies $g(X,Y)=g(JX,JY)$ for every $X,Y$ sections of the tangent ...
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Lagrangian Method for Christoffel Symbol and (non-)holonomic basis

I rencently learned about the lagrangian/variational method for computing Christoffel symbols. Let $\mathcal{M}$ be a $m-$dimensional manifold with $g_{ij}$ being the metric tensor components and ...
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291 views

Riemann sphere: Does it absolutely need two or more charts?

I am trying to understand the Riemann sphere. In this Wikipedia article it is stated that ...[the Riemann sphere] is the one-point compactification of a plane into the sphere. So it seems to ...
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Ask for an explicit proof of linear lemma in Riemannian geometry

Lemma Let $f:(M,g)\rightarrow(\bar{M},\bar{g})$ be an isometry between two Riemannian manifolds, then $df(\nabla_X Y)=\bar{\nabla}_{df(X)} df(Y)$ where $\nabla,\bar{\nabla}$ are Riemannian connections ...
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67 views

How to calculate the derivative of a Lie bracket in a coordinate-free setting?

For a given Riemmanian connection defined on a smooth manifold $M$, we denote its covariant derivative by $D_V$ where $V\in \mathcal{x}(M)$, the smooth vector fields on this manifold. Then is it ...
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48 views

Volume form for a product manifold.

Given two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$, we can construct a product Riemannian manifold $(M\times N,g^{M \times N})$ as described in Product of Riemannian manifolds? . Is there a ...
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Vanishing of the Riemann tensor

The Riemann tensor in a coordinate basis is $$R^{i}_{\,jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^m_{jl}\Gamma^i_{mk} - \Gamma^m_{jk}\Gamma^i_{ml}$$ Consider $\mathbb{R}^2$ ...
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Dimensionality of tangent vectors in R^2

I am puzzled with the following problem: given a tangent vector (a d/dx) in the Euclidean plane R^2 with "a" a dimensionless scalar, the dimensionality of this vector is, I suppose, 1/[lenght] and ...
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Does $ \int_{M} || \nabla_{M} f||^2 dS = \int_{\Omega} ||\nabla_{\mathbb{R^n}} (f \circ \Phi)||^2g d\lambda,$ hold?

Let $\Phi: \Omega \subset \mathbb{R}^n \rightarrow M$ and $M$ a euclidean manifold. Is it then correct that $$ \int_{M} || \nabla_{M} f||^2 dS = \int_{\Omega} ||\nabla_{\mathbb{R^n}} (f \circ ...
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Zero Sectional Curvature implies exp is a local isometry

Im studying DoCarmo's book Riemannian Geometry, the first problem of the chapter 5 (Jacobi Fields) states that If $(M,g)$ is a riemannian manifold with sectional curvature identically zero, show that ...
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Alternative way to recognize that a real symmetric quadratic form is positive

A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a ...
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Dimension of scalar solutions to these self-dual/anti-self-dual equations

Let $M$ be a 4-dimensional Riemannian manifold. Let $\kappa$ be a 1-form. I look for solution function $\phi$, such that there exists functions $\alpha$ and $\beta$ \begin{equation} {\left( ...
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Showing two forms on a manifold are equal

Let $\alpha$ and $\beta$ be two forms on a manifold $M$. To show that they are equal, does it suffice to show that for arbitrary $p\in M$ there exists some chart such that $\alpha_p=\beta_p$. I was ...
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40 views

How small can an external angle of a circumference be if made of tangents?

Lets imagine the angle ABC where the lines AB and CB are tangents to a circumference which center is C. Lets assume that the points where the line AB touches the circumference is P and the point where ...
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Geodesic flow on a non-complete Riemannian manifold with constant positive curvature

I am trying to understand the geodesic flow on the following 2-dimensional Riemannian manifold $M$. As a set, $M$ is the interior of the standard 2-simplex, $$M=\{(x,y)\in\mathbb{R}^2\mid ...