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 ...

learn more… | top users | synonyms

3
votes
1answer
67 views

Products of LCF manifolds

Is the Cartesian product of two Weyl-flat manifolds Weyl-flat as well? Here, by "Weyl-flat" I mean that the Weyl tensor of the metric vanishes everywhere. I know that the product of space forms isn't ...
3
votes
1answer
134 views

Computing Curvatures

What are some manifolds other than products of space forms for which the various curvature quantities can be computed easily? I'm interested in odd (real) dimensions just as much even, so I'd like to ...
1
vote
1answer
145 views

Finding a metric on a tubular neighborhood of an embedded surface

The setup for my question is an embedded surface $\Sigma\to M$ in a smooth, compact 4-manifold $M$. Assuming one knows the induced metric $g_\Sigma$ on $\Sigma$ , I would like to know if there is a ...
2
votes
1answer
358 views

Laplace-Beltrami Operator on Surfaces

I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature. For instance, What is the spectrum of the ...
2
votes
2answers
634 views

Parameterizing Geodesics on the Sphere in Polar Coordinates

I seem to have this seemingly trivial Problem, but can't figure it out. Situation: I have my Unit Sphere, $S^2$ defined as a Riemannian Manifold. Parameterizing Geodesics (Great circles) on this ...
1
vote
0answers
366 views

a question about the geodesic circle

How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature? Thanks
6
votes
1answer
564 views

Does the quotient manifold inherit the Riemannian structure?

Let $(M, g)$ be a Riemannian manifold, and let $G$ be a group acting freely and properly on $M$. From differential geometry we know that the quotient set $M/G$, i.e., the set of the orbits, is a ...
6
votes
1answer
321 views

Coordinate-free differentiation techniques in Riemannian geometry

I encountered the following identities while reading this article on global calculus (p. 10): $$ d(\|df\|^2)=2\mathop{\iota_{\mathop{\mathrm{grad}} f}} \mathop{\mathrm{Hess}} f, $$ $$ ...
1
vote
2answers
173 views

rescaled metric quantities on rescaling metrics

I have the following basic, surely stupid, questions. Assume we have a Riemannian metric $g$ on a manifold $M$. let $a\in\mathbb{R}$ a constant and consider the metric $g_1=ag$. Which are the ...
5
votes
1answer
867 views

Shape operator and principal curvature

I am trying to make geometric sense of the definition of principal curvatures as the eigenvalues of the shape operator, but I am a bit stuck. Could I have some help in showing that the principal ...
10
votes
2answers
282 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 ...
2
votes
1answer
119 views

Stitching together piece of flat space

If I start with an infinite flat sheet of graph paper, and in polar coordinates cut out a piece according to: $r>0, \ \ -f(r) < \theta < f(r)$ Now I want to stitch the remaining graph paper ...
19
votes
2answers
470 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 ...
3
votes
1answer
445 views

Levi-Civita connection of a left-invariant metric

How do I compute Levi-Civita connection of a left-invariant metric on a Lie group in a neighbourhood of $1$ by knowing only its Lie algebra and the metric form on it? I know it's possible because a ...
1
vote
1answer
159 views

Spinor Bundle on $S^{2}$

What is the spinor bundle on $S^{2}$, I mean how does it look like. Is $S$ a spinor bundle on $S^{2}$ : If $K$ be the canonical bundle, then $S = K^{1/2} \otimes \Lambda^{0,1}(S^{2})$.
7
votes
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 ...
6
votes
2answers
5k views

Proof that the angle sum of a triangle is always greater than 180 degrees in elliptic geometry

I've scoured the internet and have found many proofs showing that in Euclidean geometry, the angle sum of a triangle is always 180 degrees. I've also found many proofs showing that in hyperbolic ...
1
vote
1answer
495 views

Formula relating covariant derivative and exterior derivative

According to Gallot-Hulin-Lafontaine one has $$d\alpha (X_0,\cdots,X_q) = \sum_{i=0}^q (-1)^i D_{X_i} \alpha (X_1,\cdots,X_{i-1},X_0,X_{i+1},\cdots,X_q)$$ It seems to me that it should be $$d\alpha ...
6
votes
1answer
1k views

geodesics on a surface of revolution

I'm having problems with exercise 1 of chapter 3 of do Carmo's "Riemannian Geometry". Here is the background: Let $(u,v)$ be the coordinates on $\mathbb{R}^2$. Let $f,g\in C^\infty(\mathbb{R})$, ...
2
votes
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 ...
3
votes
1answer
295 views

Excercise in isometries of the half upper plane

This is one of Do Carmo's excersices and I got it as homework. Part (a) is easy and I include it here for the sake of completness. But I am entirely lost on part (b). A function $g:\mathbb{R} ...
0
votes
1answer
318 views

Smooth Vector Bundle is a Submersion

Let π:E→B be a smooth vector bundle. Prove π is a submersion.
11
votes
1answer
306 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 ...
7
votes
2answers
3k views

Is there a good way to compute Christoffel Symbols

Lets say you have a Riemannian Manifold $(M,g)$, and you have some given chart where $g = g_{ij} dx_i dx_j$ and you wish to compute the Christoffel symbols for the Riemannian connection in this chart. ...
1
vote
1answer
584 views

Proving some basic properties of covariant differentiation

I have the following somewhat awkward definition of covariant differentiation along a curve: Let $S \subseteq \mathbb{R}^N$ be a smoothly and isometrically embedded manifold, and $\alpha : I \to ...
5
votes
1answer
228 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 ...
2
votes
0answers
159 views

Embedding of punctured torus with euclidean metric

A flat two-torus, $T^2$ that is the torus with Euclidean metric needs to at least be embedded in $\mathbb{R}^4$. If we puncture the torus and leave the Euclidean metric on it as inherited (ignoring ...
1
vote
1answer
170 views

Completely Geodesic

I'm having trouble showing the following implication: Let $M$ be a Riemannian manifold, let $L\subset M$ be a submanifold such that the following holds: If $\gamma: I \to M$ is a geodesic s.t. ...
3
votes
1answer
465 views

Nonpositive curvature, Theorem of Cartan-Hadamard

In my differential geometry course we had the following Theorem (Cartan-Hadamard): Let $M$ be a connected, simply connected, complete Riemannian manifold. Then the following are equivalent: $M$ has ...
6
votes
1answer
408 views

Relationship between Riemannian Exponential Map and Lie Exponential Map

It is well known that for a matrix Lie group the Lie exponential map is $e ^z$. This maps a tangent vector $z$ at the identity to a group element. On the other hand the general Riemannian ...
7
votes
0answers
279 views

Riemannian Immersions into Euclidean Space?

The Whitney embedding theorem states that any smooth manifold can be embedded in Euclidean space. In the Riemannian setting this naturally leads to the question whether this can be done in such a way ...
5
votes
1answer
539 views

Quick question on Riemannian geometry

I got a quick question on riemannian geometry. I'm not quite sure whether this is the right place to ask this question, since it might be a rather elementary one from a research point of view. ...