(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

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28
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510 views

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 ...
26
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2answers
871 views

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 ...
22
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2answers
482 views

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. ...
19
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2answers
464 views

Yarn-like functions

When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than ...
14
votes
1answer
211 views

Can a continuous map $S^2 \rightarrow S^2$ preserve orthogonality without being an isometry?

Suppose I have a map $\phi: S^2 \rightarrow S^2$ and I know that a) $\phi$ is continuous and bijective b) If $a$ and $b$ subtend an angle of $\pi / 2$ at the center of the sphere, then so do ...
14
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2answers
950 views

The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
14
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1answer
304 views

Question about the proof of the index theorem appearing in Milnor's Morse Theory

I am trying to get through the proof of the index theorem. The background: I have been stuck for quite a while on the following point which Milnor says is evident: Let $\gamma: [0,1]\rightarrow M$ be ...
12
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1answer
417 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
12
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2answers
421 views

Computation of Laplace-Beltrami operator in a conformally equivalent metric

Could anyone tell me where I'm wrong with the following elementary calculation? Given a smooth Riemannian manifold $(M, g)$, I'd prove that if $\tilde{g}$ is conformally equivalent to $g$ (that is, ...
12
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2answers
438 views

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} + ...
12
votes
2answers
354 views

metric in the Wasserstein space of gaussian measures

I am reading the paper "Wasserstein Geometry of Gaussian measures" by Asuka Takatsu (section 3 is of interest to me) and I have difficulties understanding how the metric is used. In particular, I am ...
12
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1answer
406 views

Adjoint of the covariant derivative on a Riemannian manifold

Let $\nabla_X$ be the covariant derivative on a Riemannian manifold w.r.t. the vector field $X$. It is not clear to me what the (formal) adjoint of this operator is: I mean the operator ...
12
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2answers
306 views

Riemannian metric making a given function harmonic

I have a nice 3-manifold (closed, oriented) which fibers over the circle, i.e. we are given a fibration $f:M\to S^1$. Apparently $M$ should admit a metric such that $f$ is harmonic. I don't quite ...
11
votes
3answers
997 views

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, ...
11
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2answers
2k views

Definitions of Hessian in Riemannian Geometry

I am wondering is there any quick way to see the following two definitions of Hessian are coinside with each othere without using local coordinates? $\operatorname{Hess}(f)(X,Y)= \langle \nabla_X ...
11
votes
3answers
315 views

Why is $\langle \operatorname{grad} f, X\rangle_g$ independent of the metric on a Riemannian manifold?

Let $(M,g)$ be a Riemannian manifold and let $f \in C^{\infty}(M)$. Let $X$ be a smooth vector field on $M$. In smooth local coordinates $(x^i)$ on $M$, we can write $g = g_{ij} dx^i \otimes dx^j$ as ...
11
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1answer
322 views

Dolbeault Cohomology is invariant under homeomorphisms

If $X$ and $Y$ are two complex manifolds, which are homeomorphic but not necessarily diffeomorphic, must their Dolbeault cohomology groups be isomorphic? Here the Dolbeault cohomology groups ...
11
votes
1answer
301 views

Smooth Poincaré Conjecture

One of my professors wrote the following open question on the blackboard: If $M$ is a compact, connected smooth $4$-manifold such that $\pi_1(M) = 0$, $\pi_2(M) = 0$ (first two homotopy groups are ...
11
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2answers
270 views

What meaning did Riemann assign to $dx$?

Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay Uber die Hypothesen, welche der Geometrie ...
11
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1answer
198 views

why is the spectrum of the schrödinger operator discrete?

let (M,g) be a compact riemannian manifold. Then the spectrum of the Schrödinger opartor $H=-\Delta +V$ with bounded potential V acting on $L^2(M)$ consists of discrete Eigenvalues ...
10
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2answers
1k views

Is the Nash Embedding Theorem a special case of the Whitney Embedding Theorem?

The Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space. The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean ...
10
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2answers
268 views

The equivalence of two formulae for the Laplace—Beltrami operator

Let $M$ be a (pseudo-)Riemannian manifold with metric $g_{ab}$. Let $\nabla_a$ be the Levi-Civita connection on $M$. It's well-known that the Laplace—Beltrami operator can be defined in this context ...
10
votes
2answers
249 views

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from ...
10
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2answers
252 views

Are closed geodesics the prime numbers of Riemannian manifolds?

I wonder to what extent one can support the analogy that primitive closed geodesics are the prime numbers of Riemannian manifolds? ("Primitive": traced once, as opposed to $m$-fold for $m \ge 2$.) In ...
10
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2answers
367 views

Reversing the Ricci flow

Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest paths on the surface. If ...
10
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1answer
130 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
10
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2answers
294 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: $$ ...
9
votes
3answers
393 views

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 ...
9
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1answer
151 views

Is every scalar differential operator on $(M,g)$ that commutes with isometries a polynomial of the Laplacian?

On $(\mathbb{R}^n, g_{\text{std}})$ with $\Delta$ the Laplacian, the following holds: Fact: Every scalar differential operator $D$ that satisfies $D \circ F^* = F^* \circ D$ for all isometries $F ...
9
votes
1answer
195 views

Which coefficients of the characteristic polynomial of the shape operator are isometric invariants?

Let $M^n \subset \mathbb{R}^{n+1}$ be an isometrically immersed Riemannian hypersurface. The shape operator $s$ is the $(1,1)$ tensor field characterized by $$\langle X, sY \rangle = \langle ...
8
votes
4answers
695 views

Why are we interested in closed geodesics?

There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). In the case of geodesics ...
8
votes
2answers
240 views

Cartan Theorem.

Cartan Theorem: Let $M$ be a compact riemannian manifold. Let $\pi_1(M)$ be the set of all the classes of free homotopy of $M.$ Then in each non trival class there is a closed geodesic. (i.e a closed ...
8
votes
2answers
397 views

sign error proving product rule for the Laplacian on a product of Riemannian manifolds

Given two Riemannian manifolds $M$ and $N$, of dimension $m$ and $n$ respectively, the product manifold $M\times N$ has a Riemannian structure, and there is a Laplacian operator $\Delta_{M\times N}$ ...
8
votes
2answers
536 views

Why do horizontal curves have zero covariant derivative along their projection?

Background: Let $(M,g)$ be a Riemannian manifold. Let $(p,v) \in TM$ and $V, W \in T_{(p,v)}TM$. We can introduce a Riemannian metric on $TM$ via $$\langle V, W\rangle_{(p,v)} = \langle d\pi(V), ...
8
votes
1answer
175 views

What kind of polygonal surface has an interior angle > 360°?

Consider this polygon as the setting for a dynamical billiard: When it's drawn in the plane, the polygon intersects itself; it is non-simple. However, I don't want to embed the polygon in the ...
8
votes
2answers
408 views

Any compact embedded $2$-dimensional hypersurface in $\mathbb R^3$ has a point of positive Gaussian curvature

Problem statement: Let $M \subseteq \mathbb{R}^3$ be a compact, embedded, 2-dimensional Riemannian submanifold. Show that $M$ cannot have $K \leq 0$ everywhere, where $K$ stands for the Gauss ...
8
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1answer
215 views

Creating geodesics on manifolds

Suppose I have two points on a Riemannian manifold $M$, called $p_0$ and $p_1$. I have a family of curves $\gamma:[0,\infty)\times[0,L]\to M$ such that $\gamma(t,0) = p_0$ and $\gamma(t,L) = p_1$. ...
8
votes
2answers
273 views

Total mean curvature of an immersed torus.

How to prove that the total mean curvature of an immersed torus of $R^3$ such that has nontrivial self-intersection must $> 8 \pi$? The definition of total mean curvature is the integral of $H^2$ ...
8
votes
1answer
2k views

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 ...
8
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1answer
99 views

Using index notation to write $d^2=0$ in terms of a torsion free connection.

Let $(M,g)$ be a Riemannian manifold and let $\omega$ be a $1$-form on $M$. I want to rewrite $d^2\omega=0$ in terms of the Levi-Civita connection. I can show the following: $$d\omega(X,Y) = ...
8
votes
1answer
81 views

Are there more embeddings $U(2) \hookrightarrow SO(4)$?

It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$. My ...
7
votes
4answers
294 views

Is length adimensional when space is not flat?

Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...
7
votes
3answers
279 views

Are the Ricci and Scalar curvatures the only “interesting” tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. ...
7
votes
1answer
318 views

Most succinct proof of the uniqueness and existence of the Levi-Civita connection.

Seeing as proving the existence and/or uniqueness of the Levi-Civita connection seems to crop up in every single exam in Geometry and General Relativity, what is the most succinct proof of this, to ...
7
votes
2answers
437 views

Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

Let $(M,g)$ be a Riemannian manifold. Then I want to show that these are equivalent: (i) Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal. ...
7
votes
2answers
341 views

About Gauss-Bonnet Theorem

The Gauss–Bonnet theorem say that: If $\Sigma \subset M =\mathbb{R}^3$ is a compact 2-dimensional Riemannian manifold without boundary, then $$ \int_{\Sigma} K = 2\pi\chi_{\Sigma}$$ where $K$ is the ...
7
votes
1answer
129 views

Is there a codifferential for a covariant exterior derivative?

For forms on a Riemannian $n$-manifold $(M,g)$ there is a notion of a codifferential $\delta$, which is adjoint to the exterior derivative: $$\int \langle d \alpha, \beta \rangle \operatorname{vol} = ...
7
votes
1answer
968 views

Isometries preserve geodesics

Let $f$ be an isometry (i.e a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds $(M,g)$ and $(N,h).$ One can argue that $f$ also preserves the induced metrics $d_1, ...
7
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2answers
197 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 ...
7
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1answer
1k views

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 ...