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I don't know much about infinite dimensional things. I am not sure about the right answer but may be following may be useful... Once i saw the following book and statement: See Page 4 of the book "Le …

Dear Sir/friends, How to give manifold structure to set of all $C^2$ path over any manifold.

$M$ be any Riemannian manifold, and $S^1$ is a circle. We can give Manifold structure to $C^\infty(S^1, M)$ modeled on nuclear frechet space. Take $Imm(S^1, M):\{f\in C^\infty(S^1,M): f \text{ is an …

"Suppose $N$ is $2n-1; n\geq 2$ dimensional $CR$ manifold and everywhere Levi flat, then it will be locally $CR$ equivalent to $S^1\times \mathbb C^{n-1}.$" Above statement can be found in Loop sp …

We know that for any simply connected surface $M$,whose Gaussian curvature $K\leq 0$, for any $p\in M$, $exp_p: T_pM\to M$ is diffeomorphism. We know that for any $v\in T_pM$ and $w\in T_v(T_pM)\sim …

If $v=0$, then we have $F(v)= G(v)=0$ Hence obviously we have $F(v)= \lambda. G(v)$ for some non zero real $\lambda$. Now if $v\neq 0$, then for any $g\in GL_n(\mathbb R)$, we have $g(v)\neq 0$ and …

Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ext …

Let $(U,u)$ is a chart for $M$, and $(V,v)$ be a chart for $N$. $u: U\to u(U)\mathbb R^n$ is diffeomorphism. $u(U)$ and $v(V)$ are open subset of $\mathbb R^n$ and $\mathbb R^m$. Then we can identify …

Hello, $CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point. I can't see what the happening in Non …