Intereting Posts

Jump Continuity
Projection of tetrahedron to complex plane
Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$
Matrix Exponential does not map open balls to open balls?
Find the Vectorial Equation of the intersection between surfaces $f(x,y) = x^2 + y^2$ and $g(x,y) = xy + 10$
Paracompact image of an open continuous map from a Čech-complete space
Value of cyclotomic polynomial evaluated at 1
Integrating $\int_0^\pi \frac{x\cos x}{1+\sin^2 x}dx$
Complex Conjugate of Complex function
How to solve $\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$ with condition $X(0)=1$?
How to obtain the series of the common elementary functions without using derivatives?
Extended Euclidean algorithm with negative numbers
Intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?
Proof that the area under a curve is the definite integral, without the fundamental theorem of calculus
expected value of a function

Let $\pi: E \to M$ be a smooth vector bundle of rank $n$, and suppose $s_1, \ldots, s_m$ are independent smooth local sections over an open subset $U \subset M$.

Can I prove the “basis extension theorem” for smooth sections? That is, for each $p \in U$, there are smooth sections $s_{m+1},\ldots,s_n$ defined on some neighborhood of $V$ of $p$ such that $(s_1, \ldots, s_n)$ is a smooth local frame for $E$ over $U \cap V$.

- how to compute the de Rham cohomology of the punctured plane just by Calculus?
- Intuition for Smooth Manifolds
- Differentiable injective function betweem manifolds
- Number of Differentiable Structures on a Smooth Manifold
- Push-forward of vector fields by local isometries
- Space of smooth structures

- Second Stiefel-Whitney Class of a 3 Manifold
- $\Bbb{R}P^1$ bundle isomorphic to the Moebius bundle
- Is there a retraction of a non-orientable manifold to its boundary?
- The tangent bundle of a Lie group is trivial
- parallelizable manifolds
- Understanding the Definition of a Differential Form of Degree $k$
- Extension of vector bundles on $\mathbb{CP}^1$
- How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?
- Quotient of $\textrm{GL}(2,\textbf{R})$ by the conjugate action of $\textrm{SO}(2,\textbf{R})$
- Reference request: infinite-dimensional manifolds

Yes. Let $p:E\to M$ denote the fibration. Let $W\to U$ be the subbundle of $p^{-1}(U)\to U$ given by $W=\coprod _{p\in U}span\{s_{1}(p),…,s_{m}(p)\}$. Since the secions are smooth, this does indeed define a subbundle. Now let me give you an idea of how to proceed. You want to look at the normal bundle. If you fix a metric $g$ on $M$, there is a natural choice of such a bundle. Namely you can define $N=\coprod_{p\in U}N_{p}$ where $N_{p}$ is the orthogonal complement of $W_{p}$. Check (using the metric) that this is indeed a vector bundle. Then $N$ must be trivial, since $U$ is contractible. Choose a global section of the frame bundle of $N$ and your done!. You can do this without fixing a metric on $M$ too, you just need to use the quotient bundle $p^{-1}(U)/W$.

Let $W$ be a neighborhood of $p$ in $M$ such that there is a local trivialization $\varPhi:\pi^{-1}(W)\to W\times \mathbb{R}^n$. Then $v_i(q)=(\pi_{\mathbb{R^n}}\circ\varPhi\circ s_i)(q)$ are linearly independent in $\mathbb{R^n}$ for each $q\in W$. We can find $v_m, \ldots,v_n \in \mathbb{R}$ such that $\{v_1(p), \ldots,v_m(p),v_{m+1},\ldots,v_n\}$ is a basis for $\mathbb{R^n}$. So $\det [v_1(p), \ldots,v_m(p),v_{m+1},\ldots,v_n]\neq0$, because the $\det$ is continuous there is a neighborhood $V$ of $p$ such that $\det [v_1(q), \ldots,v_m(q),v_{m+1},\ldots,v_n]\neq0$ for each $q\in V$. Then $(s_1, \ldots s_m,s_{m+1}=\varPhi^{-1}(\cdot,v_{m+1}), \ldots, s_{n}=\varPhi^{-1}(\cdot,v_{n})) $ is a frame over $U\cap V $.

- Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?
- Quick method for finding eigenvalues and eigenvectors in a symmetric $5 \times 5$ matrix?
- Integral $\int_0^1 \log \frac{1+ax}{1-ax}\frac{dx}{x\sqrt{1-x^2}}=\pi\arcsin a$
- Localization of a ring which is not a domain
- Finding the range of $f(x) = 1/((x-1)(x-2))$
- Are finite indecomposable groups necessarily simple?
- Possible mistake in Folland real analysis?
- Subgroups of finite abelian groups.
- C* algebra inequalities
- Show that the tangent space of the diagonal is the diagonal of the product of tangent space
- Communication complexity example problem
- Group of order $48$ must have a normal subgroup of order $8$ or $16$
- Eigenvalues of an operator
- Topologies in a Riemannian Manifold
- Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$