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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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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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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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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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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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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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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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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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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33 views

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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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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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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49 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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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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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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51 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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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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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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86 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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160 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. ...
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Sum of Killing vector fields is a Killing vector field

Let $(M,g)$ be a Riemannian manifold. A smooth vector field $X$ is called a Killing vector field if the flow of $X$ acts by isometries, or, equivalently, if $L_X g = 0$. Now why is the sum of Killing ...
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Compute a parallel transport

Let $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ be the $2$-sphere ($\mathbb{S}^{2} = \left\{ (x,y,z) \in \mathbb{R}^3, \; x^2+y^2+z^2 = 1 \right\}$). Let $p \in \mathbb{S}^{2}$ and $\xi \in T_{p}S^{2} = ...
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What is Model Spaces

I am reading the Riemannian Geometry, written by Lee, and have just finished the Chapter 3, which ends with The Model Spaces of Riemannian Geometry. There are three kinds of model spaces $\mathbb ...
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53 views

Isometries are affine transformations

I want to show that, if $(M,\mathrm{g})$ is a Riemannian manifold, $\nabla$ is the covariant derivative from the Levi-Civita connection, and $f:M\to M$ is an isometry, then ...
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72 views

A question regarding Jacobi fields and families of geodesics

I'm trying to show that for any one-parameter family of geodesics $\gamma(s,t)$ (where $\gamma(s_0,t)$ is a geodesic for any constant $s_0 \in (-\epsilon, \epsilon)$) defined on a Riemannian manifold ...
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Identity involving Riemann tensor

I'm reading about the Ricci tensor, and I've found the following statement that is given without proof: For a point $p$ on a Riemannian manifold, and coordinate vector fields $X_{\alpha}$, ...
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93 views

In what sense is a pseudo-Riemannian metric a “metric”?

I have read that Riemannian manifolds have the structure of a metric space. In this sense, they have a distance function and it satisfies the definition of metric space. However, I have recently ...
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Invariance of determinant of metric tensor

Given any 2-tensor on a Riemannian manifold $M$ equipped with metric $g,$ we have a coordinate-free definition of its trace: $$\operatorname{trace}(T)=g^{ij}T_{ij}= T_i^i.$$ In particular, we have ...
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What significant differences are there between a Riemannian manifold and a pseudo-Riemannian manifold?

I am reading John Lee's book Riemannian Manifolds. On page 91, he begins a chapter called "Geodesics and Distance," which is I think the first chapter that seriously addresses geodesics. I was very ...
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80 views

Constant curvature metrics on the sphere

Are there Riemannian metrics other than the standard metric induced from the euclidean space on $S^2$ such that the sectional curvature is equal to 1 everywhere? Or is this the unique Riemannian ...
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Negative Gauss Curvature

Let S be a manifold of dimension 2, compact and orientable. Suppose its border is made of k geodesic circumferences, with $k \geq 3$. Show that there exists a point in S with negative Gauss ...
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Fundamental group of cusp of a negatively curved manifold

Let $M$ be a complete, noncompact Riemannian manifold with finite volume and whose sectional curvatures vary within the interval $[a,b]$, $-1\leq a<b<0$. It is known that such manifold has ends ...
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One parameter subgroups on Lie groups and Riemannian metric

I read that geodesics of a bi-invariant metric on a compact Lie group are the one parameter subgroups. In a general Lie group, is it possible to create a Riemannian metric by transporting the ...
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Proof a $(2n-1)$-compact manifold

I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold. How give the ...
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How does an affine connection permit differentiation of vector fields?

As I understand it, one primary use of affine connections is to "connect" tangent spaces. Suppose I take a velocity vector $\dot{\gamma}(t_0)$ on a curve and at some point $\dot{\gamma}(t)$ also on ...
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Does existance of a Killing spinor imply existance of a Killing vector?

I am wondering about the relationship between Killing spinors and Killing vector fields. In the nlab entry for Killing spinors the quote "Pairing two covariant constant spinors to a vector yields a ...
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54 views

Rotation by $90°$ in differential geometry

Let $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a parametrized surface and $\nabla_{c'}c'$ be the covariant derivative of a curve $c:I \rightarrow \Omega$ that is parametrized by ...
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179 views

Parametrised vs Regular Surfaces

Two types of surfaces in $\mathbb{R}^3$ are usually studied in introductory books on differential geometry: Parametrised or immersed surface: Is an immersion $F:U\rightarrow\mathbb{R}^3$ from an ...
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91 views

Continuous function with non-negative second derivative in the weak sense is convex

I am currently working through a section of Peter Petersen's Riemannian Geometry in which he talks about weak second derivatives of functions. I am trying to work through the details of why a function ...
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Help understanding John Lee's definition of curvature

On page 3 of his book Riemannian Manifolds, John Lee states the following If you want to continue your study of plane geometry beyond figures constructed from lines and circles, sooner or later you ...
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Characterisation of local affine diffeomorphisms

I've got a question about local affine diffeomorphisms between affine manifolds. There ist a good characterisation about affine diffeomorphisms of connected affine Mannifolds: Let $f,g\colon ...
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46 views

Non-constant rank of a smooth map and orthnormal basis in the normal bundle

Assume $M,N$ are two Riemannian manifolds and $f: M\rightarrow N$ is a smooth map. Suppose $dim M =m < dim N =n$. Let $\Sigma$ be the graph of $f$, that is, $\Sigma =(x, f(x))$ for $x\in M$. My ...
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Equivalent definitions of a surface

do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post. Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or ...
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Finding a frame for a vector bundle in a smooth manifold with a connection

I am trying to solve the following exercise: Let $P$ be a vector bundle over a smooth manifold $M$ with a connection $\nabla$, and let $p \in M$ . Show that there is an open set $U$ of $M$ with $p ...
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Is the tangent-cotangent isomorphism orientation preserving?

Consider $(M,g)$ a Riemannian manifold. Let's define $\varphi : TM\rightarrow T^{\ast}M$ by $\varphi(p,v):=(p,g(v,.))$, for $p\in M$ and $v\in T_{p}M$. Here, $TM$ stands for the tangent bundle and ...
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What's wrong in this prop about volume form if we drop “oriented”?

I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption. I know that I'd came up with a non zero ...
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Does a Riemannian metric allow definition of a tangent vector's length?

In Euclidean spaces, we define the Euclidean norm of a vector $\vec{x} = (x_1,x_2,...x_n)$ as $\|\vec{x}\|:=\sqrt{x_1^2+x_2^2+ \cdots +x_n^2 }$ Does the metric tensor field of a Riemannian manifold ...
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40 views

How can I measure the length of a curve using a Riemannian metric?

On page 35 in his book Riemannian Geometry, Manfredo do Carmo states the following: Giving a surface $S \subset \Bbb{R}^{3}$, we have a natural way of measuring the lengths of vectors tangent to ...
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Is $M \times (0,\infty)$ a manifold of bounded geometry?

If $M$ is a compact Riemannian manifold, is $M \times (0,\infty)$ a manifold of bounded geometry? I think it is, since $M$ is compact and $(0,\infty)$ is simply flat.