Questions tagged [connections]

Question 1

Consider $\mathbb{R}^n$ with its standard inner product and usual affine connection $\nabla^{\mathrm{eucl}}$. Let $M$ be a manifold and suppose that one is given an immersion $f:M\to\mathbb{R}^n$. At ...

Question 2

Given a connection on the tangent bundle $TM$ of $M$, how can we define an exterior derivative of a vector valued (with values in a vector bundle) $k$-form? Thank you for your answers.

Question 3

Let $E$ be a rank 2 complex vector bundle over a 4-dimensional manifold $X$. (I believe that the argument below does not depend on the rank of $E$ and the dimension of $X$.) I want to try to show, for ...

Question 4

Let $M$ be a $n$-dimensional Riemannian manifold, i.e. $M = \cup_{\lambda \in \Lambda}\, (U_{\lambda}, g_{\lambda})$ with each $U_{\lambda}$ being an open ball around the origin of ${\Bbb R}^n$ and $...

Question 5

There is an old issue I couldn't fix by myself. I consider a manifold $M$, with frame bundle $E$, a flat and torsionfree affine connection $\nabla_0$ and an affine connection $\nabla$. I consider them ...

Question 6

I'm reading a classical paper by Katz and Oda on the Gauss-Manin connections, and I have question on one of its claims. Let $S$ be a smooth $k$-scheme and $\mathscr{E} \in \operatorname{QCoh}(S)$. Let ...

Question 7

Let $E$ denote an elliptic curve over $\mathbb{C}$. I would like to consider meromorphic connections on the trivial line bundle $O_E^{\oplus r}$ with regular singularities occuring at some finite set ...

Question 8

Let ${\mathrm{S}}^1$ be a circle with the radius $\frac{1}{2}$, which is a one-dimensional Riemannian manifold. I would like to give the chart on ${\mathrm{S}}^1$ by the affine $x$-real line and the ...

Question 9

Let $P_{\Bbb C}^1$ be a projective line over the complex number field ${\Bbb C}$. We have $P_{\Bbb C}^1 \thickapprox S_{\Bbb R}^2$ as the Riemannian manifolds. There is the Levi-Civita connection on $...

Question 10

There are two definitions of the affine connection on the tangent bundle of a $\mathcal{C}^\infty$ manifold $M$. One being in terms of a Differential Operator i.e. Definition: An Affine Connection is ...

Question 11

In singularity theory, one defines an intrinsic derivative for a vector bunle homomorphism $\phi: E \rightarrow F$ where $E\xrightarrow{\pi_E}B$ and $F\xrightarrow{\pi_F}B$ are vector bundles with ...

Question 12

In Section 10.2 of Loring W. Tu’s Differential Geometry: Connections, Curvature, and Characteristic Classes, he explains how to patch connections on a vector bundle using partitions of unity. He uses ...

Question 13

I am trying to find an intuitive interpretation to the Riemann tensor in the presence of torsion. If we look at the index definition: \begin{equation} x^by^cR^a{}_{ bcd}z^d=x^by^c\left[\nabla_b,\...

Question 14

This is about Nakahara's "Geometry, Topology and Physics", second edition, proposition 10.2. I am not understanding the simple statement at the very beginning of the proof. $$ (R_{a*} X)^H = ...

Question 15

I'm working on problem 4-11 of Lee's Introduction to Riemannian manifolds,here is it: Suppose $G$ is a Lie group. (a) Show that there is a unique connection $\nabla$ in $TG$ with the property that ...

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Questions tagged [connections]

Do affine connections always come from immersions in $\mathbb{R}^n$?

Exterior covariant derivative of a vector valued $k$-form [closed]

Showing that the trace of the difference of the square of the curvatures of two connections on a vector bundle is exact

Why is the connection necessary?

Recovering the formula for curvature from matrix form using Cartan formalism

Flat connections and Lie-algebra homomorphisms from Der to End

Meromorphic Connections on an Elliptic Curve

Levi-Civita connection for ${\mathrm{S}}^1$.

Line bundle and the ‘‘algebraic’’ Levi-Civita connection.

Why are these two notions of Affine Connections equivalent?

Intrinsic derivative

Patching connections with partitions of unity preserves torsion?

Riemann Tensor with Torsion Intuition

Question about proof of proposition 10.2 in Nakahara's book $(R_{a} X)^H = R_{a}(X^H)$

problem about connection in Lie group(solved)

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