
Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" used to describe geometric data. All sheaf data in the LRS approach can be described by bundles u...

Recently, the following tag synonyms were created by a MathOveflow moderator: abstract-algebra $\to$ ra.rings-and-algebras, geometry $\to$ mg.metric-geometry and discrete-mathematics $\to$ co.combinatorics. You can see the new synonyms in the list of tag synonyms (Wayback Machine). The three tags...

De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the levi-Chevita connection is isometric to the direct product of two Riemanninan manifolds $M'\times...

Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$.
Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$
be a generic fiber that is a submanifold of $F$.
Assume the singular fibers are $F/\Gamma_t$, where for each $t\in Y\setminus U$, ...
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