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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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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25 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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37 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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Linearisation in direction of formal adjoint

Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. The Ricci curvature can be viewed as a differential operator ...
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42 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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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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32 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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metric reversed on product manifolds

metric reversed is in section 3 of semi riemanian geometry Oniell. I need some examples with details of metric reversed on product manifolds.
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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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56 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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58 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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33 views

harmonic function on manifold

Let M be a 2 dimensional manifold. $h:M\rightarrow R$ be a harmonic function from manifold to real line. G is group that act by isometry. $g*h(x)=h(g(x))$. Let $W=\{x|h(x)=t\}$ that is the level set ...
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Recommendable books to study the Selberg zeta function.

I've study on the Riemann zeta function and some zeta functions which have analytic properties directly. And now I want to know about the Selberg's zeta function which has some geometric properties. ...
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77 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" ...
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40 views

Schwarzschild half-plane and its geodesics

For some fixed $r_0>0,$ put the semi-Riemannian metric $$ ds^2=\frac{r_0-r}{r}dt^2+\frac{r}{r-r_0}dr^2 $$ on $\{(t,r)\in\mathbb{R}^2:r>r_0\}.$ I would like to show that the $r$-lines are always ...
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20 views

What exactly are the independent components?

What are the 20 independent, non zero components of the 4D Riemann curvature tensor? (Not how many, I know there are twenty, but specifically which components are non-zero?)
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65 views

Verification of product rule for covariant derivatives. Stuck on one step involving simplifying terms to yield zero.

I am trying to learn more about covariant differentiation. I'm specifically interested in physics applications, but I found this nice exercise in Misner, Thorne, and Wheeler's book Gravitation that I ...
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32 views

Projection of fiber bundle is a submersion

I'm just wondering about my proof for the following fact. I get the feeling it is almost trivial but I am still getting a feel for geometry and so it doesn't seem 'obvious' to me just yet. The ...
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60 views

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions:

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions: a. There is another point $p_{0}$ such that the ...
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54 views

definition of derivative of a vector field

i try to define by myself the notion of differentiation of a vector field on a general manifold. I know that it is a classical subject and that there exist some answers as Lie derivative of a vector ...
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1answer
38 views

Riemann manifold with unbounded Laplacian

How can one characterize a Riemann manifold the Laplacian of which is unbounded? (Equivalently, what are those manifolds on which the Laplacian is bounded? I am interested in working with its ...
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217 views

Equivalence of two distance function on a Riemannian manifold

Let $(M,g)$ be a closed connected $m$ dimensional smooth Riemannian manifold and assume that it is isometrically embedded in a Euclidean space $\mathbb{R}^q$ by $\iota:M\to\mathbb{R}^q$. $|\ast|$ ...
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Prove properties of induced connection from a Levi-Civita connection

I just learnt let $M=$Riemannian manifold,$N=$differentiable manifold, $\phi:N\to M$ be smooth map. If $v\in T_xM$, and $\{E_i\}_{i=1}^n$ is a frame field in a neighborhood $V$ of $\phi(x)\in M$, ...
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37 views

What does $\langle R(x,z_i)x,z_i \rangle$ represent geometrically?

In his book Riemannian Geometry, Manfredo Do Carmo states the following on page 97: Let $x = z_n$ be a unit vector in $T_pM$; we take an orthonormal basis $\lbrace z_1,z_2,...,z_{n-1}\rbrace$ of ...
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47 views

Computation of the extrinsic curvature tensor for a warp drive metric.

In Miguel Alcubierre's renowned paper discussing a "warp drive" metric, he discusses the extrinsic curvature. Here is an extract. My questions are quite trivial to someone who understands the ...
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Induced connection is well-defined

Let $M=$Riemannian manifold,$N=$differentiable manifold, $\phi:N\to M$ be smooth map. If $v\in T_xM$, and $\{E_i\}_{i=1}^n$ is a frame field in a neighborhood $V$ of $\phi(x)\in M$, then $$\forall ...
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1answer
25 views

$D_vX$ is completely determined by $X$ restricted on any curve $r$ with $r'(0)=v,r(0)=p$.

I want to show given $v\in T_pM$, then $D_vX$ is completely determined by $X$ restricted to any curve $r$ with $r'(0)=v,r(0)=p$. I have shown that if $r_1'(0)=r'_2(0)=v,$ then ...
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16 views

Recognizing regular distributions

By "regular" distributions I understand those Schwartz distributions that arise from locally-integrable functions. Are there ways of telling them apart from the non-regular ones? Does the set of those ...
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2answers
87 views

Equivalence for Christoffel symbol and Koszul formula

I am trying to show to define a Levi-civita connection, it's equivalent to define Christoffel symbols or define Koszul formula. $$ 2g(\nabla_XY, Z) = \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - ...
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1answer
31 views

An explicit Lorentzian metric on the Klein bottle

I want to construct an explicit Lorentzian metric on the (abstract) Klein bottle but have no idea where to start. Could someone please give me a hint and/or guide me in the right direction? Thanks.
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58 views

Dehn twist as isometries on hyperbolic surface

[I am editing the question to a most correct and precise one thanks to comments of Lor and studiosus] Let (S,g) be a compact hyperbolic surface. On a simple closed geodesic $\gamma $ I can realized a ...
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54 views

hyperbolic geometry and affine transformations

A very famous geometer recently commented to me that "hyperbolic geometries are the only geometries invariant under affine transformations". It is unclear to me what this comment even means. Can ...
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2answers
81 views

Can someone explain the basic idea behind the sectional curvature formula?

I found the following equation on Wikipedia here: \begin{equation} K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2} \end{equation} No explanation I ...
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1answer
46 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
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79 views

Visualizing Ricci scalar curvature

I am trying to learn more about Ricci scalar curvature. I am trying to get an image in my head of what scalar curvature actually represents about the curvature of a manifold. The most familiar image I ...
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1answer
28 views

Flat connection of a vector bundle over a 1 dim. manifold

I'd like to show that a connection of a vector bundle $E$ over a 1 dim. manifold $M$ is flat, or equiv. that its curvature is zero. Let $D$ denote the connection, $\sigma$ a section of $E$ and $v,w$ ...
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Interpolation of the metric tensor

I am currently facing the following problem. I have a Riemannian manifold, where the metric is only known at certain points. Are there some standard strategy to interpolate the metric in other points ...
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1answer
39 views

Question about curvature calculation method in Lee's *Riemannian Manifolds* book

In his book Riemannian manifolds, John Lee states the following on pages 8-9: The most fundamental fact about geodesics is that given any point $p\in M$ and any vector $V$ tangent to $M$ at $p$, ...
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Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
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66 views

Check Christoffel symbol defines Levi-Civita connection

I am trying to prove the existence of Levi-Civita connection. The hint says given $(U_\beta,\phi_\beta)$ be altas of $M$, for $X=x^i\partial_i,V=v^j\partial_j$, we define $$D_VX=v^i(\partial_i ...
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1answer
33 views

How to prove that volume forms agree on $U_\alpha \cap U_\beta$?

I am familiarizing myself with Riemannian manifolds. Let $M$ be an orientable smooth $n$-manifold with atlas $(U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$ and let $g$ be a Riemannian metric on $M$. ...
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1answer
40 views

Do we write a metric tensor as a matrix?

The metric tensor is an (0,2) tensor that is denoted by $g_{\mu\nu}$ in general relativity. I often see people write the metric field in matrix form like \begin{equation} g_{\mu\nu} = ...
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1answer
49 views

Can I argue like this to prove that the determinant is positive?

Let $X$ be a smooth $n$-manifold with an oriented atlas $\mathcal U = (U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$. Let $g$ be a Riemannian metric on $X$. Let $g_{ij} = g\left ( {\partial \over ...
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191 views

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
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44 views

existence of affine connection on manifold

I am studying Riemannian Geometry following my professor's notes. On the proof of existence of affine connection on a $C^\infty$ manifold, the notes states: By partition of unity, a connection can be ...
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57 views

Why is the trace of the Riemann curvature tensor useful?

As I understand it, the Ricci curvature tensor is the trace of the Riemann curvature tensor. In other words, \begin{equation} R_{ij} := R_{kij}^{\phantom{kij}k} = g^{km}R_{kijm} \end{equation} But ...
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66 views

Clarification of definition of tensor product

I am reading "Riemannian Geometry" by Gallot. And I am confused with the following definition of tensor product: Let $E$ and $F$ are two finite dimensional vector spaces, a vector space $E\otimes ...
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1answer
77 views

Pullback of a differential form

My question is in regards to a proof in Lee's 'Introduction to Smooth Manifolds'. He proves a lemma about the pullback of a differential form on a manifold $N$, where $F:M\rightarrow N$ is a smooth ...
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2answers
146 views

Hessian of a function on Riemannian manifolds

Let $(M,g,\nabla)$ be a Riemannian manifold with metric $g$ and Riemannian connection $\nabla$. The hessian of a function $f:M\to R$ is defined by: $$H^f(X,Y)=g(\nabla_X\ \ \operatorname{grad} ...
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Is there a better way to show the intrinsic curvature of a cylinder is zero?

I am new to differential geometry and Riemannian geometry. I have on two separate occasions (separated by 6 months) encountered exercises where I feel like I am not giving a complete answer. ...