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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A slice orthogonal to each orbit

Assume that a compact (connected) Lie group $G$ acts on a manifold $M$. We choose a $G$-invariant Riemannian metric on $M$ and a point $p \in M$. Then using the exponential map at $p$, we can obtain a ...
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Identites of Riemann curvature tensor on orthonormal frame

Suppose we consider a Riemannian manifold and a local orthonormal frame $\{Y_i\}$. I was wondering whether, for the Riemann curvature tensor $R$, there are identities with regards to expressions of ...
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On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
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367 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
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Riemannian Manifolds with $n(n+1)/2$ dimensional symmetry group

Given a $n$-dimensional connected Riemannian manifold $(M,g)$, its symmetry group $G$ can be considered as a subbundle of orthonormal frame bundle of $M$ (which I call $F_OM$), yielding: $$\dim G\le ...
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Geodesics Examples

Can someone provide me exemples of connected Riemannian manifolds containing two points through each there are : (i) infinitely many geodesics (up to reparametrization) and (ii) no geodesics. ...
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113 views

The cut-off function on Riemannian manifold

Let $M$ be a complete Riemannian manifold. Can we find a constant $C>0$ so that for any $p\in M$ and $R>0, 2R<\text{inj}(M)$, we can find a function $\varphi \in C^\infty _c(B_p(2R))$, such ...
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74 views

Existence of geodesics and sign of the gaussian curvature

This is Problem 11 of this year's Miklos Schweitzer. (a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to ...
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126 views

Relation between exponential map and parallel transport

I'm starting to learn Riemannian geometry and have a question. Let $\mathcal{M}$ be a Riemannian manifold, $p \in \mathcal{M};\ \tau_{p}^{q}$ be a parallel transport from $p$ to $q$ and ...
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381 views

Homogeneous riemannian manifolds are complete. Trouble understanding proof.

I came across this proof while looking for hints on my homework, and I think it's only gotten me more confused. This is from Global Lorentzian Geometry. Lemma 5.4 If $(H,h)$ is a Riemannian ...
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325 views

Ricci curvature: step in proof of a paper by Hamilton

In Hamilton's paper "The Ricci Curvature Equation" (in Seminar on Nonlinear Partial Differential Equations, here), I can do all of Lemma 4.2 except for the following relation: $$ ...
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72 views

Why are geodesics preserved by the quotient with the isometry group $M/G$?

I'm trying to prove that if $\langle M,g\rangle$ is a riemannian manifold and $G = Isom(M)$ acts properly discontinuous on $M$, then a geodesic $c$ is send to another geodesic by the map $\pi: M ...
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149 views

Regularized distance function on Riemannian manifolds

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let ...
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309 views

Tensors constructed out of metric other than the Riemann curvature tensor

Let $(M,g_{ab})$ be a Riemannian (or Pseudo-Riemannian) manifold and let us define a tensor field as 'something' that transforms in an appropriate way under coordinate transformations. (This is how ...
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154 views

Diffeomorphism invariant scalars of a Riemannian manifold

Let $(M,g_{ab})$ be a Riemannian manifold. I know of the following scalars that one can construct them out of the metric and its derivatives: Ricci scalar $R$ $R_{ab}R^{ab}$ $R_{abcd}R^{abcd}$ ...
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75 views

Is set of focal points of a submanifold on a normal geodesic discrete?

Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...
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72 views

The control of geodesic rays

Let $M$ be a simply-connected, complete Riemannian manifold whose sectional curvature $K_M$ satisfies $-b^2\leq K_M\leq -a^2<0$. Fix two points $p,q$ in $M$, for any geodesic ray $\gamma(t)$ ...
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367 views

Negative curvature compact manifolds

I know there is a theorem about the existence of metrics with constant negative curvature in compact orientable surfaces with genus greater than 1. My intuition of the meaning of genus make me think ...
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Geodesics on pseudo-Riemannian manifolds

Consider a Riemannian manifold $M$, with a metric $g$. We can find univocally the Levi Civita connection $\nabla$ on $M$, and so a covariant derivative $D_t$ (associated to $\nabla$) along curves. A ...
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geometric intent behind defining definiteness of scalar product.

I would like to know how the definiteness/semi-definiteness(positive or negative)of a scalar product (i.e. a symmetric bilinear form) influence the geometric nature of a space on which they are ...
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141 views

Shape Operators and Symmetric Linear Transformations

The exercise (from Sakai) is: Let $f: E\subseteq \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ be smooth and let $M_f := \{p = (x, f(x)) \in \mathbb{R}^n\,;\,x \in E\}$ be the graph of $f$ considered ...
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Trouble understanding Tensor product in context of Torsion Tensor

I know that there are quite a few threads dealing with this question already. I have pored through them for quite some time and they have been informative. However there are still some clarifications ...
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87 views

About the symmetry of Riemann Tensor

It is a problem in my homework. First I was asked to show $$ \nabla_a\nabla_bA_c-\nabla_b\nabla_aA_c=R_{a,b,c}^{\;\;\;\;\;d}A_d $$ where $A$ is a (0,1)-tensor and $R_{a,b,c}^{\;\;\;\;\;d}$ is the ...
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69 views

Show that $\operatorname{div} X = - \delta X^\flat$

I want to show the equality $\operatorname{div} X = -\delta X^\flat$, where $X \in \Gamma(TM)$ and $M$ is some Riemannian manifold with metric tensor $g_{ij}$. If I'm not mistaken it holds for the ...
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158 views

Problem with notation: Laplacian on a manifold

In the Aubin's book "Nonlinear analysis on manifolds" the Laplacian operator on functions on some smooth manifold is defined by the formula $$ \Delta = -\nabla^\gamma\nabla_\gamma, $$ where ...
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48 views

Gaussian Curvature of the half-plane only with the metric

Let $S= \{(u,v): u \gt 0 \}$ the half-plane. And the metric $$g=du^2+u^2dv^2$$ How to find the Gaussian curvature of S? I don't if there is sufficiently information to do this.
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219 views

Hessian matrix on Riemannian manifolds

$(M, g)$ is a Riemannian manifold, and $u$ is a smooth function on $M$, one says that under a normal coordinate system, $\operatorname{Hess}(u)_{ij} = (u_{lk} - u_h \Gamma^h _{lk})B^{li}B^{kj}$, where ...
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109 views

negative Euler characteristic $\Rightarrow$ homotopy unique up to homotopy

In a paper by John Franks I stumbled upon the following: Let $M$ be a surface and $f:M \rightarrow M$ be a homeomorphism, which is homotopic to the identity on $M$. That means, that there is ...
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84 views

quotient of 2-torus by antiholomorphic involution is annulus?

I would like to study what the quotient $$T^2 / \Omega $$ of a closed compact Riemann surface with $g=1$ handles, once a complex structure is chosen, over an antiholomorphic involution $\Omega,$ can ...
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122 views

Linear connection on a manifold: Math vs. Physics

I have learned some Riemannian Geometry in a strongly mathematical framework, precisely from the book "J.M.Lee - Riemannian Manifolds: An introduction to Curvature". Now I'm trying to learn ...
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106 views

Show that $a \wedge * b = g(a,b) \operatorname{vol}$

$\newcommand{\vol}{\operatorname{vol}}$ Let $\omega$ be a $p$-form on a Riemannian manifold $M^n$ with metric $g$ and let $\vol_{i_1,\ldots,i_n}=\sqrt{\lvert g\rvert} \epsilon_{i_1,\ldots,i_n}$ be a ...
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Suggestion for a good book that explain Cartan's Moving Frame and Riemannian Geometry

I'm studying Riemannian Geometry, and I'm having a lot of trouble with the book Riemannian Geometry and Differential Dorms both from do Carmo.And I would like a book with examples, calculations, if ...
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Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
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Proving that Hermitian Metric yields Hermitian Structure on Complex Manifold

Let $g$ be a Riemannian metric on an almost complex manifold $(M,J)$. Suppose $g$ is Hermitian in the sense that $$g(JX,JY) = g(X,Y)$$ Let $\Omega$ be the associated fundamental (Kahler) form ...
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How to find a dual frame at the sphere

Let $U=\{ (\theta,\phi,r):\theta \in \mathbb{R}, \phi \in ]0,\pi[,r\gt 0\}$, how I can find the moving frame. I thougt: Consider the parametization for $U$ $$(\theta,\phi,r)\mapsto ...
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156 views

Distance between a point and the origin in Poincaré Disk

How to calculate the distance between the origin and the point $p=(a,0)$, $a>0$, using the metric $g = \frac{4}{(1-x^2-y^2)^2}(dx^2+dy^2)$? I don't how to use correctly these $dx^2$'s
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Laplacian on a warped product.

Let $(M, g)$, $(N, h)$ be complete Riemannian manifolds (not necessarily compact). Let $f : M \rightarrow (0, \infty)$ be a smooth function, and finally let $$\overline{M} = M \times_f N$$ be the ...
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Can there be non trivial self-dual 5-forms on a 10-dimensional compact orientable manifold without boundary?

I am puzzled about the following. Let $(M,g)$ be a compact, orientable Riemannian manifold without boundary. We define the usual inner product $(,)$ for two r-forms $\alpha,\beta\in\Omega^r(M)$ by ...
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122 views

Harmonic function on Riemannian manifold

$M$ is a connected Riemannian manifold with $\Delta$ its Laplacian and $f$ is smooth function on $M$ such that $\Delta f=0$ and $f$ vanishes on some open set $U$ of $M$, then is $f$ identically $0$ on ...
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46 views

The choice of the eigenfunction of Laplacian

$M$ is a closed Riemannian manifold and $\lambda_1>0$ is the first nontrivial eigenvalue of $\Delta$. Can we find a eigenfunction $f$ of $\lambda_1$ such that $\mathop {\sup }\limits_M f - \mathop ...
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42 views

Are part of great circles geodesics on spherically symmetric manifolds?

While doing some study on geodesics on Riemannian manifolds, I learned that any geodesics on $n$-spheres are part of great circles. I then started wondering if that is true for any spherically ...
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236 views

Bianchi identity of linear connection on vector bundle

Consider a connection on $E$ which is a vector bundle over $M$ : $$ \nabla : \Gamma(E) \rightarrow \Omega^1(M)\otimes \Gamma(E),\ s\mapsto \nabla\ s$$ Here $\nabla s =dx^k\otimes ...
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194 views

derivative of a positive definite matrix

Suppose that $A$ is a positive definite symmetric matrix, specifically a Riemannian metric. Can we say anything about the sign of $tr(A^{-1}\partial_i A)$?
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Jacobi fields in polar coordinates.

This is from Sakai's Riemannian Geometry: Let $(r, \theta)$ be polar coordinates of the plane. We define a Riemannian metric $g$ on the plane by $g(\frac{\partial}{\partial r}, ...
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366 views

Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
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203 views

Euler Lagrange equation for harmonic maps

In the paper "The existence of minimal immersions of 2-spheres" by Sacks and Uhlenbeck the authors claim that the Euler Lagrange equation for the modified functional $E_\alpha(s) = \int_M (1 + ...
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68 views

$2$-dimensional Hyperbolic space with fundamental group ${\bf Z}$ and constant curvature $-1$

$$ d\rho^2 + \cosh^2\rho\ d\theta^2$$ Only one ? Is there any other example ?
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Orthogonal irreducible decomposition of $\otimes^2 E$

Recall $$ \otimes^2 E = \wedge^2 E \oplus S^2_0E\oplus {\bf R}$$ Clearly this is $O(n)$-decomposition. Irreducibility can be checked from the following property : Let $Ae_1=e_k,\ Ae_2=e_l,\ ...
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Decomposition of $S^2(\wedge^2 E)$

Consider bianchi map $$ b(T)(x,y,z,t) = \frac{1}{3}(T(x,y,z,t)+T(y,z,x,t) + T(z,x,y,t))$$ where $T\in S^2(\wedge^2 E)$ I already checked that $b(b(T))=b(T)\in S^2(\wedge^2 E)$ But how can we derive ...
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70 views

Fundamental group of a component of $GL_n({\bf R})$

Let $G$ be a component of $GL_n({\bf R})$ such that element has a positive determenant. (1) Since it contains $SO(n)$, $\pi_1(SO(n))$ ? What is a fundamental group of $G$ ? (2) It has a curvature ...