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Results tagged with connections
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user 46290
Ehresmann connections; covariant derivatives; connections on vector bundles, principal bundles, ∞-bundles, submersions, bundle gerbes; holonomy and higher holonomy; parallel transport; torsion; curvature. See also the tags [principal-bundles], [vector-bundles], [gerbes], [curvature], [geodesics], [characteristic-classes], [torsion].
2
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1
answer
232
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If any two triangles of equal area can be mapped via affine maps, what can we say about the ...
This is a cross-post.
Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric.
Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there exists …
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5
votes
1
answer
253
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Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$?
This is a cross-post.
Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\nabla$, …
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0
votes
Accepted
Does "symmetry" of a pullback connection should be obvious?
$\newcommand{\id}{\operatorname{Id}}$
Well, there is a natural way to view this "pullback-symmetry":
Exterior derivative commutes with pullbacks:
Let $f:M \to N$ be a smooth map, $E$ a vector bundl …
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1
vote
1
answer
135
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How large can the cone of $\nabla$-compatible metrics be?
Let $E$ be a smooth vector bundle over a manifold $M$, equipped with a connection $\nabla$.
The set of $\nabla$-compatible metrics on $E$ forms a convex cone.
This cone can be empty, however (see he …
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2
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2
answers
645
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Does "symmetry" of a pullback connection should be obvious?
$\newcommand{\M}{M}$
$\newcommand{\N}{N}$
$\newcommand{\TM}{TM}$
$\newcommand{\TN}{TN}$
$\newcommand{\TstarM}{T^*M}$
$\newcommand{\Ga}{\Gamma}$
Let $\M,\N$ be smooth manifolds, $\phi:\M \to \N$ be a …
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