(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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185 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind

I need to prove the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ ...
6
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1answer
954 views

Conformal transformation of the curvature and related quantities

Suppose we have a Riemannian manifold ${(M,g)}$, where ${g}$ is the metric of ${M}$. If ${f}$ ${\in}$ ${D(M)}$ (i.e. smooth function on ${M}$), and ${f}$ is positive. So, we can define a new metric ...
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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 ...
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1answer
419 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. ...
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3answers
830 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, ...
6
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1answer
634 views

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 ...
14
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2answers
879 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 ...
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2answers
2k views

Isometries of the sphere $\mathbb{S}^{n}$

Got this as homework and I don't know how to tackle this. Help please! Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to ...
2
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1answer
205 views

Tensors constructed out of metric other than the Riemann curvature tensor

Let $(M,g_{ab})$ be a Riemannian (or Pseudo-Riemannian) manifold and let us define a tensor field as 'something' that transforms in an appropriate way under coordinate transformations. (This is how ...
5
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1answer
206 views

Extension of Riemannian Metric to Higher Forms

I've been reading about Riemannian manifolds, and have come across a comment that says that for a metric $g$ on an $N$-dimensional manifold $M$, considered as a bilinear map $$ g:\Omega^1(M) \times ...
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2answers
254 views

The punctured unit disc has the complete riemannian metric with constant curvature -1

Find how to construct this metric, find the distance under the metric between $(e^{-2\pi},0)$ and $(-e^{-\pi},0)$ This is a very interesting question, I have an idea ,construct Riemannian covering ...
3
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1answer
135 views

Need help finding a good book on Riemann Geometry

I want to learn more about calculus on manifolds and Riemann Geometry. I have been reading the book Geometry, Topology and Physics by Nakahara. But I find that it is difficult to read due to the lack ...
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1answer
391 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 ...
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1answer
1k 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 ...
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1answer
927 views

Riemannian metric of the tangent bundle

I'm trying to solve the following problem (from do Carmo's Riemannian Geometry). particularly I'm having trouble proving that the inner product defined is bilinear. Problem. It is possible to define ...
4
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1answer
89 views

Relation between two Riemannain connections

Let $g$ be a Riemannian metric on $M$ and let $\tilde{g}=f^{2}g$ where $f$ is a smooth function that is never zero. let $\nabla$ and $\nabla'$ be the Riemannain connections of $g$ and $\tilde{g}$ on ...
9
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1answer
180 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 ...
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votes
2answers
152 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 ...
6
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1answer
145 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
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votes
2answers
424 views

Geodesics on the product of manifolds

Given two Riemannian manifolds $(M, g_1)$ and $(N, g_2)$, and geodesic curves $\gamma(t)$ in $M$ and $\chi(t)$ in $N$. Is the curve $\Gamma(t) = (\gamma(t),\chi(t))$ a geodesic in the product manifold ...
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votes
2answers
645 views

Riemannian Geometry book to complement General Relativity course?

What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure ...
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0answers
101 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
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1answer
291 views

Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
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votes
2answers
382 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 ...
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292 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
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1answer
167 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
5
votes
2answers
188 views

Compatibility of a connection and metric

Every Riemannian manifold admits a metric connection. Suppose $M $ is a manifold and $\nabla $ is an arbitrary connection on the tangent bundle. Does $M$ necessarily admit a metric such that $\nabla$ ...
5
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3answers
397 views

Geodesics of a “diagonal” metric

Are there any relations that exist to simplify Christoffel symbols/connection coefficients for a diagonal metric which has the same function of the coordinates at each entry? In other words, I have a ...
5
votes
3answers
836 views

Expression of the Hyperbolic Distance in the Upper Half Plane

While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia. ...
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279 views

Does the curvature determine the metric?

Here I asked the question whether the curvature deterined the metric. Since I am unfortunately completely new to Riemannian geometry, I wanted to ask, if somebody could give and explain a concrete ...
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2answers
586 views

Riemannian metric in the projective space

Let $A: \mathbb{S}^n \rightarrow \mathbb{S}^n$ be the antipode map ($A(p)=-p$) it is easy to see that $A$ is a isometry, how to use this fact to induce a riemannian metric in the projective space such ...
3
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1answer
228 views

Berger's theorem on holonomy

Can someone clarify to me what the correct hypothesis of Berger's theorem are (if at all what I write is correct)? Theorem: assume $M$ is a Riemannian manifold, with irreducible reduced holonomy ...
2
votes
1answer
136 views

Gradient of a functional

Given a compact manifold with a Riemannian metric $g$, we define the total scalar curvature by $$E(g)=\int_M RdV$$ Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
2
votes
1answer
419 views

Quotient theorem for tensors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
11
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1answer
314 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 ...
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votes
3answers
344 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 ...
7
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1answer
319 views

Positive definiteness of Fubini-Study metric

Define the Fubini-Study metric $$g_{i\overline{j}} = \frac{\delta_{i\overline{j}}(1+|\boldsymbol{z}|^2)-\overline{z}^jz^i}{(1+|\boldsymbol{z}|^2)^2} $$ for $i,j=1,\ldots,n$ and $z_i$ complex variables ...
5
votes
2answers
271 views

Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
5
votes
2answers
248 views

Justification for this manipulation in a proof of the first variation of energy formula

As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in these ...
5
votes
2answers
112 views

Equivalence of intrinsic and extrinsic metrics of embedded manifolds.

Say a compact n-manifold $\mathcal{M}$ is embedded in $\mathbb{R}^m$, $m > n$. If $d_{\mathcal{M}}$ is the geodesic distance on $\mathcal{M}$, and $d$ the Euclidean distance in $\mathbb{R}^m$, ...
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0answers
95 views

Diffeomorphism invariant scalars of a Riemannian manifold

Let $(M,g_{ab})$ be a Riemannian manifold. I know of the following scalars that one can construct them out of the metric and its derivatives: Ricci scalar $R$ $R_{ab}R^{ab}$ $R_{abcd}R^{abcd}$ ...
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votes
3answers
297 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kähler condition implies that $\omega$ is closed with respect to $d$, so ...
4
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2answers
262 views

Moving to a conformal metric

Given a generic 2-dimensional metric $$ ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2 $$ what is the change of coordinates that move it into the conformal form $$ ...
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1answer
521 views

Fixed Points Set of an Isometry

I'm reading Kobayashi's "Transformation Groups In Riemannian Geometry". I'm trying to understand the proof of the following theorem: Theorem. Let $M$ be a Riemannian manifold and $K$ any set of ...
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1answer
122 views

What is the importance of conformal vector fields on Riemannian manifolds?

A vector $X$ on a Riemannian manifold $(M,g)$ is called conformal if $L_{X}(g)=2sg$ where $L_{x}$ is the Lie derivative and $s$ is a real-valued function on $M$. If $s=0$, $X$ is called a killing ...
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77 views

Critical paths for length cannot have kinks.

This problem is in Spivak's Differential Geometry (Ch.9 #37), and he gives a sketch of a proof which I have been unable to finish. So let's compute $\frac{dL(\overline{\alpha}(u))}{du}\mid_{u=0}$ ...
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0answers
217 views

Does the curvature determine the metric for all surfaces

In order to not make things even more confusing than they are, I split my two-in-one question into two parts. Here's the second part (the first part is here): Here I asked the question whether the ...
2
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1answer
104 views

The degree of Gauss map

If $M$ is an $2m$-dimensional closed orientable hypersurface in $\mathbb R^{2m+1}$, then we have a Gauss map $G:M\rightarrow S^{2m}$. I have known from my differential geometry book that ...
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1answer
305 views

The behavior of all unit speed geodesics on a surface of revolution.

In the $xz$-plane of $\mathbb R^3$, consider the closed non-singular curve $\gamma$ which is the image of the function $$t\mapsto (1+2\sin^2(t))(\cos(t),0,\sin(t)).$$ (Note that $\gamma$ is invariant ...
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1answer
72 views

Autoparallel submanifolds and geodesics

I have the following question in differential geometry. Any help is greatly appreciated. Let $M$ be an autoparallel submanifold of a manifold $S$ with respect to a connection $\nabla$. Let $\gamma$ be ...