(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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462 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 ...
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1answer
416 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 ...
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1answer
157 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 $S = K^{1/2} \otimes \Lambda^{0,1}(S^{2})$.
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1answer
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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
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2answers
4k 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 ...
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1answer
479 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
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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 $f,g\in C^\infty(\mathbb{R})$, ...
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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 ...
3
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1answer
269 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
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1answer
304 views

Smooth Vector Bundle is a Submersion

Let π:E→B be a smooth vector bundle. Prove π is a submersion.
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0answers
163 views

Are Lie Groups Homogeneous Spaces?

Is any Lie Group a homogeneous space?
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1answer
295 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
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2answers
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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. ...
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1answer
539 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
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1answer
207 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 ...
1
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1answer
166 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
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1answer
431 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
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1answer
385 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
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0answers
265 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 ...
4
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1answer
517 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. ...