Intereting Posts

Higher homology group of Eilenberg-Maclane space is trivial
Asymptotics of terms and errors in Stirling's Approximation
Prove Uncountable set minus a countable set is uncountable
Some case when the central limit theorem fails
Isomorphisms between Normed Spaces
Functions of bounded variation on all $\mathbb{R}$
Entropy of matrix
Partitions of the odd integers
How to tell if a Rubik's cube is solvable
Given any $10$ consecutive positive integers , does there exist one integer which is relatively prime to the product of the rest ?
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
Does $\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0 $ for all $n$ imply that $\lambda_1= \lambda_2= \dots= \lambda_k = 0 $?
Can you determine from the minors if the presented module is free?
Can a polygon with minimal perimeter self-intersect?
line equidistant from two sets in the plane

Why fiber of $(P\times V)/G\rightarrow P/G$ isomorphic to $V$ ?

I think the fiber should be $V/G$, but it is not isomorphic to $V$

Picture below is from the 66 page of Jost’s Riemannian Geometry and Geometric Analysis

- Equivalence of Definitions of Principal $G$-bundle
- Are there vector bundles that are not locally trivial?
- Fiber bundle with null-homotopic fiber inclusion
- Universal property of universal bundles.
- The sections of the projection $\bigsqcup_{i:I} X_i \rightarrow I.$
- Elementary proof of the fact that any orientable 3-manifold is parallelizable

- The sections of the projection $\bigsqcup_{i:I} X_i \rightarrow I.$
- Universal Cover of $SL_{2}(\mathbb{R})$
- Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?
- Is this reasoning correct? Connection with torsion on SO(3)
- One-dimensional Lie algebra with non-trivial bracket operation
- How to understand Weyl chambers?
- Center of $\mathfrak{sl}(n,F)$
- Complete reducibility of sl(3,F) as an sl(2,F)-module
- Meaning of vanishing Lie bracket
- Lie algebra-like structure corresponding to noncrystallographic root systems

Fix a point $x$ in the base. Then the fiber over $x$ consists of the orbits of all pairs $(p,v)\in P\times V$ such that $p$ lies in the fiber over $x$. Now fix a point $p_0$ in that fiber and consider the map from $V$ to the fiber of $P\times_GV$ over $x$ which sends $v$ to the orbit of $(p_0,v)$. This is injective since $p_0\cdot g=p_0$ implies $g=e$, so $(p_0,v)$ and $(p_0,w)$ lying in the same orbit implies $v=w$. On the other hand, for each $p$ in the fiber, there is an element $g\in G$ such that $p=p_0\cdot g$. Hence the orbit of $(p,v)$ contains $(p_0,g\cdot v)$ which implies surjectivity.

- Pullback and Pushforward Isomorphism of Sheaves
- How much set theory and logic should typical algebraists/analysts/geometers know? (soft-question)
- Why rationalize the denominator?
- Examples of measurable and non measurable functions
- How come that two inductive subsets can be different
- limit of $\left( 1-\frac{1}{n}\right)^{n}$
- The total ring of fractions of a reduced Noetherian ring is a direct product of fields
- Definition of Equivalent Norms
- Equicontinuity on a compact metric space turns pointwise to uniform convergence
- Prove that the limit of $\sin n$ as $n \rightarrow \infty$ does not exist
- What are the epimorphisms in the category of Hausdorff spaces?
- Does Stirling's formula give the correct number of digits for $n!\phantom{}$?
- Totally disconnected space
- Probability distribution and their related distributions
- How is this subgroup a normal subgroup?