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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Question about parallel displacement on a surface

This is Problem 9.6(1) from the book The Geometry of Physics: What's wrong with the following argument? A vector $\mathbf v$ is parallel displaced around a small closed curve $C = ...
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28 views

Unbounded Geodesics and Nonpositive Curvature

I have the following interesting(?) question: Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics. As the question is ...
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50 views

product metric on Riemannian manifolds

Let $M_1$ and $M_2$ be Riemannian manifolds and consider the cartesian product $M_1 \times M_2$ with the profuct structure .Let $\pi_1: M_1 \times M_2 \to M_1$ and $\pi_2: M_1 \times M_2 \to M_2$be ...
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24 views

Existence of a particular vector in kernal of $d \exp_x$

I want to prove $(d\exp_x)_{p}$ is singular iff there exists a normal Jacobi field $U(t)$ along $\gamma(t)=\exp_x(tp)$ not identically zero such that $U(0)=U(1)=0$. I have question about ...
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44 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
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93 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
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50 views

Different definitions of tangent vector

I'm taking general relativity at the moment, and today in class the instructor gave us a definition of tangent vector as: $v$ is a tangent vector based at $p\in M$ if $v_{p}$ is a linear ...
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42 views

To calculate covariant derivative in Riemanian Geometry

I want to solve the exercise $2.57$ using $2.56$ I know calculate $2.57$ by using christoffel symbols but this process is long.How can I solve this directly via $2.56$. Can somebody help me by ...
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40 views

How do you get the curvature tensor of the Schwarzschild Solution?

So, on the Wikipedia page on the derivation of the Schwärzschild solution , I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
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21 views

CAT(K) Finsler manifolds.

I was wondering if the following is true (and common knowledge): Let $(M,F)$ be a Finsler manifold. Let d be the induced distance by the norm in the usual sense. That is, $d(x,y)=\inf${lenghts of ...
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27 views

A triangle inequality for angles

Let $M$ be a complete Riemannian manifold with nonnegative curvature and $x,y,z,p$ four points on $M$. We denote by $\theta(x,y),\theta(y,z),\theta(z,x)$, respectively, the angles at $\tilde p$ of the ...
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61 views

Gradient in local coordinates on a manifold with Riemannian metric

Let $M$ be a smooth manifold with a Riemannian metric g : $TM\otimes TM$ -> R If f is a smooth function from M to R, the gradient of f with respect to g is the vector field $\nabla f$ defined by ...
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60 views

Why is $\frac{d}{dt}|\xi(t)|^2=2<\xi, \nabla \xi>$, when $\nabla$ is the covariant derivative?

I understand that: $(1) \frac{d}{dt}|\xi(t)|^2=2<\xi, \nabla \xi>$ if this is happening in $\mathbb{R}^n$. But my question is in the following setting: $\nabla \xi:=K_{c^*\pi} \circ T ...
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50 views

Exterior differentiation of one form on a smooth manifold

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I'm fine with the right side of the equation, ...
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34 views

Converse of statement related to Hopf-Rinow Theorem

I know the Hopf-Rinow theorem and that if a Riemannian manifold is complete it implies that given any two points there is a unique distance minimizing geodesic that connects the two points, but is the ...
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50 views

Extension of isometries on submanifolds of a riemannian manifold

Let $S$ a submanifold of a riemannian manifold $M$ such that the closure of $S$ is equal $M$. i.e $\bar{S}=M$, when can we extend an isometry(as riemannian manifold) $f:S\to S$ to an isometry ...
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48 views

Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and ...
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75 views

Coordinate expression for the divergent

Let $(M, g)$ be a Riemannian manifold. As in Lee's Riemannian Manifolds book, we define the divergent of a vector field $X \in \mathfrak{X}(M)$ by the identity $d(\iota_X dV) = (div X) dV$, where ...
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56 views

Expression for codifferential in terms of interior product

Let $(M^n,g)$ be a Riemannian manifold with local orthonormal frame $\{e_1,\ldots,e_n\}$ with dual basis $\{e^1,\ldots,e^n\}$ and with Levi-Civita connection $\nabla$. It can be checked on basis that ...
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30 views

a question about finding umbilical points in an elipsoid.

Determine the umbilical points of the elipsoid $${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$$ My thoughts: let $x=asin(\theta)cos(\phi),y=bsin(\theta)cos(\phi),$and $z=cos(\theta)$. Thus, I ...
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a question about differential geometry

Let $S$ be a surface and $x: U\to S$ be a parametrization of $S$. If $ac-b^2 <0$, show that $$a(u,v)(\dot u)^2+2b(u,v)\dot u\dot v+c(u,v)(\dot v)^2=0$$ can be factored into two distinct equations, ...
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133 views

Sets that are convex in two different metrics

Let $(M,g)$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining ...
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40 views

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result: I have three questions: (1) Is $h$ also a conformally flat ...
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20 views

metric reversed on $S(n,v)$

On page 204 of O'Neill, Barrett (1983), Semi-Riemanniann geometry warped product is explained. How can I calculate the metric tensor on $S(n,v)$ where $v$ is the index of $S$ and $n$ is its ...
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24 views

All rank two symmetric tensors are several conformally flat metrics summed together?

If given a rank two symmetric tensor $T_{mn}$ can I decompose it as \begin{align} T_{mn} = \sum_{i=1}^{M} \phi_ig_{mn}{}^{i}{} \end{align} where $\phi_i$ are the $i$th conformal factors and ...
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53 views

Notation in symmetric and alternating products of forms

In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric ...
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67 views

$\Delta_L(\text{im}\,\delta^*_g)\subset\text{im}\,\delta^*_g$ and $\Delta_L\big(\text{ker}\,\text{Bian}(g)\big)\subset\text{ker}\,\text{Bian}(g)$?

Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The Lichnerowicz Laplacian of $g$ is the map ...
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32 views

How to prove the skew symmetry of the Riemannian tensor?

I parallel transported a vector around a parallelogram formed by another two vectors to get the Riemannian tensor: $$R^m{}_{lkj} = \left( \partial_k \Gamma^m{}_{jl} + \Gamma^m{}_{kn} \Gamma^n{}_{jl} ...
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51 views

Kähler–Einstein condition

Let $(M,\omega)$ be a Kähler manifold, and $g$ the Riemannian metric such that $\omega(X,JY)=g(X,Y)$. If there is a function $f$ such that $\operatorname{Ric}=f\omega$ does this mean that ...
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29 views

Dilating a curved ball

Let $B$ be a ball sitting inside a manifold $(M^n, g)$. Now, let us dilate the metric $g$ to $\lambda g$, $\lambda$ being a positive number going to $\infty$. It seems intuitively true that the ...
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113 views

Adjoint of Exponential Map

If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like? ...
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What is the geometric interpretation of the Koszul formula?

I saw this simple form of Koszul formula on a book: $$2\ g(\nabla_XY,Z) = \mathcal{L}_Yg(X,Z) + (d\theta_Y)(X,Z)$$ where $\theta_Y$ is the one-form $g(Y,\cdot)$. It is equivalent to the more ...
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63 views

Lower bound on convexity radius in terms of injectivity radius (without using curvature)

Let $M$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining $p$ ...
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54 views

Differential Geometry - Distributions mutually orthogonal, span the tangent space, parallel imply manifold splits locally as product manifold

I'm stuck on a portion of Exercise 21, Chapter 2 in Petersen's Riemannian geometry text. Fix a Riemannian manifold $(M,g).$ Suppose that I have two distributions $D^1$ and $D^2$ defined on $M.$ ...
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57 views

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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73 views

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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78 views

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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64 views

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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1answer
54 views

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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35 views

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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99 views

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 ...