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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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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42 views

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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2answers
58 views

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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1answer
80 views

A determinant coming out from the computation of a volume form

I am convinced that the following identity is true: \begin{equation} \det\begin{bmatrix} 1+a_1^2 & a_1 a_2 & a_1 a_3 & \ldots & a_1a_n \\ a_1a_2 & 1+a_2^2 & a_2a_3 & ...
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77 views

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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1answer
90 views

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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32 views

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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1answer
68 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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92 views

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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57 views

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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93 views

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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22 views

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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1answer
171 views

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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147 views

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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32 views

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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1answer
39 views

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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65 views

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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2answers
340 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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1answer
63 views

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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1answer
130 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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1answer
150 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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63 views

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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30 views

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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50 views

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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260 views

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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60 views

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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26 views

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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1answer
68 views

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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1answer
47 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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52 views

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 ...
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39 views

Uniform convergence of the harmonic form heat flow

[${\bf NOTATIONS}$] Let $M$ be a closed Riemannian manifold of $m$ dimensional, $p\in\{1,\cdots,m\}$. $A^p:=\{\text{smooth p-forms on }M\}$. $\delta:A^{p+1}\to A^p$ denotes the formally adjoint ...
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1answer
254 views

Riemannian metric induced by metric

This seems a very basic and useful construction, and yet I cannot find any reference for it. So my questions are, 1) Is the following definition correct? 2) Is there a simpler construction? 3) Do you ...
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1answer
53 views

Two isometries that have same value and differential at some point are the same.

I also have trouble in this problem: Let $f, g$ be two isometries of a connected Riemannian manifold $(M, g)$. If $f(p)=g(p)$, $df_p = dg_p$, show that $f=g$. Any comment is expected. I know it ...
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1answer
32 views

Minimal geodesic on the real projective $n$-space

I have encountered this problem: Show that a geodesic of $(\mathbb RP_n, g_0)$ with $g_0$ being the metric given by the canonical metric on $\mathbb S^n$ via the $2:1$ Riemainnian covering, is ...
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1answer
53 views

Frame acting on a curve/Geodesic eqution

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
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1answer
20 views

Length shortening Riemannian metrics

I am looking for examples of Riemannian metrics such that the curve length under these metrics are always smaller than the length as measured in Euclidean space. It is just a question that popped into ...
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1answer
116 views

Exterior derivative of forms derived from a metric

Let $(M,g)$ be a Riemannian manifold. From $g$ and a fixed vector field $V$ we can derive the following two differential forms: A $1$- form $\alpha(X) = g(V,X)$, i.e. $\alpha = \iota_Vg$. A $2$-form ...
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1answer
50 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
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81 views

Coding for a calculation in differential geometry using Maple

I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple. I have these vector fields $e1=z^2\ast ...
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46 views

Is the Riemann curvature tensor the only tensor that can be constructed from the metric tensor and its first and second derivatives?

I am reading Gravitation and Cosmology by Steven Weinberg. On page 133, he says $R^{\lambda}_{\phantom{x}\mu\nu\kappa}$ is the only tensor that can be constructed from the metric tensor and its ...
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36 views

Stable geodesics

Consider a function defined on some space of smooth curves in a manifold (think of the "action functional"). I understand what a "critical point" of such a function is, but what is understood by a ...
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37 views

The Jacobian of the exponential along a geodesic

I am reading a paper that uses but does not define the following concept: what is understood by "the Jacobian of the exponential map along a geodesic (beetween two points)"? Is this only defined for ...
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1answer
88 views

Dot product of two cross products in $\Bbb R^3$ with general metric

I would like to find the generalized formula of the identity $$(A\times B).(C\times D)=(A\cdot C)(B\cdot D)-(A\cdot D)(B\cdot C)$$ which holds in an Euclidian metric, within a general metric $g$ on ...
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1answer
54 views

Two transformation groups of the hyperbolic plane are isomorphic?

I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane. I've just been told that $SO(2,1)$, the indefinite ...
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131 views

Does covariant derivative commute with “generalized contraction”_ About the proof of 2nd Bianchi identity

I am reading the proof of second Bianchi identity on wiki. In the proof, it says the following condition must satisfy: $$((D_X R) (Y,Z)) (W) + R (D_XY,Z) W + R(Y,D_XZ) W + R(Y,Z) D_X W = D_X ...
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1answer
56 views

Prove the local expression of Riemannian curvature tensor

I try to prove the following expression of Riemannian curvature tensor: For local coordinate $\{x^i\}$, let $g_{ij}=g(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})$ and ...
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179 views

Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
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1answer
60 views

Riemannian distance induced by an elliptic differential operator?

consider a Riemannian manifold $(M,g)$ and consider a second order elliptic differential operator. I've read that each such operator induces a riemannian distance function. Unfortunately I couldn't ...
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2answers
84 views

If we don't need a Riemannian metric to compare length of vectors, why do we use metrics to measure curvature?

I read that, in the absence of a Riemannian metric tensor field, we can still measure how much a vector changes when parallel transported around a curve by comparing the initial and final vectors. ...
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1answer
370 views

How are “scalar curvature” and “sectional curvature” related?

I was browsing wikipedia and was puzzeling about what is the difference between: "scalar curvature" https://en.wikipedia.org/wiki/Scalar_curvature and "sectional curvature" ...