Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

Introduction

Metaphorically, Riemannian geometry is what happens when we try to generalize the Pythagorean theorem to work on smooth manifolds in general, but accidently drop the Pythagorean theorem in a blender along the way.

Definition 1.1a: (Riemannian Metric) Suppose $M$ is a smooth manifold. A Riemannian metric on $M$ is a section $\mathrm{g}\in\Gamma(T^\vee\hspace{-.25em}M\otimes T^\vee\hspace{-.25em}M)$ such that, for each $p\in M$ and all $X_p,Y_p\in T_pM$,

  • $\mathrm{g}_p(X_p\otimes Y_p)=\mathrm{g}_p(Y_p\otimes X_p)$,

  • $\mathrm{g}_p(X_p\otimes X_p)\geq0$, with equality if and only if $X_p=0$.

Note that many mathematicians use the following equivalent definition.

Definition 1.1b: (Riemannian Metric) Suppose $M$ is a smooth manifold. A Riemannian metric is a smooth function $\mathrm{g}:TM\times_MTM\to\mathbb{R}$, where $TM\times_MTM$ is the fiber product, such that, for each $p\in M$, all $X_p,Y_p,Z_p\in T_pM$, and all $a,b\in\mathbb{R}$,

  • $\mathrm{g}(aX_p+bY_p,Z_p)=a\mathrm{g}(X_p,Z_p)+b\mathrm{g}(Y_p,Z_p)$,
  • $\mathrm{g}(X_p,Y_p)=\mathrm{g}(Y_p,X_p)$,
  • $\mathrm{g}(X_p,X_p)\geq0$, with equality if and only if $X_p=0$.

Regardless of the particulars of the definition, a Riemannian metric is essentially a smooth choice of inner product on each tangent space. Making a choice of Riemannian metric gives us a Riemannian manifold.

Definition 1.2: (Riemannian Manifold) A Riemannian manifold is a pair $(M,\mathrm{g})$, where $M$ is a smooth manifold and $\mathrm{g}$ is a Riemannian metric.

There is a plethora of examples of Riemannian manifolds that appear all over geometry.

Example 1.3: (Euclidean Space) Let $x$ be the (global) identity chart on $\mathbb{R}^n$. A Euclidean space is a Riemannian manifold of the form $$\left(\mathbb{R}^n,\sum_{i=1}^n\mathrm{d}x^i\otimes\mathrm{d}x^i\right).$$ Usually, we identify $n$-dimensional Euclidean space with $\mathbb{R}^n$.

Example 1.4: (Hyperbolic Plane) Let $(x,y)$ be the (global) identity chart on the upper half plane. Then, the hyperbolic plane is the Riemannian manifold $$H^2=\left(\mathbb{R}\times\mathbb{R}_+,\frac{1}{y^2}\left(\mathrm{d}x\otimes\mathrm{d}x+\mathrm{d}y\otimes\mathrm{d}y\right)\right).$$

This tag is for questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag if your question involves Riemannian manifolds or objects generally associated with them, such as Levi-Civita connections.

6762 questions
113
votes
1 answer

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the derivative…
58
votes
3 answers

Geometrical interpretation of Ricci curvature

I see the scalar curvature $R$ as an indicator of how a manifold curves locally (the easiest example is for a $2$-dimensional manifold $M$, where the $R=0$ in a point means that it is flat there, $R>0$ that it makes like a hill and $R<0$ that it is…
55
votes
3 answers

Definitions of Hessian in Riemannian Geometry

I am wondering if there is any quick way to see the following two definitions of Hessian coincide with each other without using local coordinates. $\operatorname{Hess}(f)(X,Y)= \langle \nabla_X \operatorname{grad}f,Y \ \rangle$;…
user17150
  • 897
  • 1
  • 8
  • 10
55
votes
4 answers

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. Fundamentally, I would like to know why the Laplacian…
54
votes
1 answer

Are there simple examples of Riemannian manifolds with zero curvature and nonzero torsion

I am trying to grasp the Riemann curvature tensor, the torsion tensor and their relationship. In particular, I'm interested in necessary and sufficient conditions for local isometry with Euclidean space (I'm talking about isometry of an open set -…
43
votes
4 answers

Existence of a Riemannian metric inducing a given distance.

Let $M$ be a smooth, finite-dimensional manifold. Suppose $M$ is also a metric space, with a given distance function $d: M \times M \rightarrow \mathbb{R}_{+}$, which is compatible with the original (manifold) topology on $M$. Question: is there a…
student
  • 3,727
  • 30
  • 38
40
votes
2 answers

Geometric meaning of symmetric connection

If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following conditions: 1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0 2) Symmetricity, i.e.…
evgeny
  • 3,653
  • 16
  • 38
40
votes
1 answer

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $C^\infty(M)$ to be those functions which have all…
37
votes
2 answers

What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $n-1$ angular coordinates) would be preferable.…
35
votes
4 answers

When does a space admit a flat metric?

Once upon a time I was told that the torus is flat. This was supposed to be surprising, since the ordinary picture of a torus we have in our heads looks inherently curved. However, thinking instead of a torus as a square in the plane with opposite…
34
votes
1 answer

Intuition for curvature in Riemannian geometry

Studying the various notion of curvature, I have not been able to get the intuition and deeper understanding beyond their definitions. Let me first give the definitions I know. Throughout, I will consider a $m-$dimensional Riemannian manifold…
32
votes
3 answers

Can every curve on a Riemannian manifold be interpreted as a geodesic of a given metric?

Given a metric $g_{\mu\nu}$ it is possible to find the equations of the geodesic on the Riemannian manifold $M$ defined by the metric itself: $$\frac{d^2x^a}{ds^2} + \Gamma^{a}_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds} = 0$$ where: $$\Gamma^a_{bc} =…
Riccardo.Alestra
  • 10,172
  • 4
  • 35
  • 71
31
votes
4 answers

Existence of a local geodesic frame

Let $(M,g)$ be a Riemannian manifold of dimension $n$ with Riemannian connection $\nabla,$ and let $p \in M.$ Show that there exists a neighborhood $U \subset M$ of $p$ and $n$ (smooth) vector fields $E_1,...,E_n \in \chi (U),$ orthonormal at each…
Ehsan M. Kermani
  • 8,851
  • 2
  • 32
  • 53
30
votes
4 answers

the Levi-Civita connection on a product of Riemannian manifolds

I'm working on exercise 1(a) of chapter 6 in do Carmo's Riemannian Geometry: Let $M_1$ and $M_2$ be Riemannian manifolds, and consider the product $M_1\times M_2$, with the product metric. Let $\nabla^1$ be the Riemannian connection of $M_1$ and…
Aaron Mazel-Gee
  • 7,749
  • 1
  • 38
  • 60
30
votes
2 answers

PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality for $H^1_0$ hold, or like the norms $|\Delta \cdot…
1
2 3
99 100