On the diffetential of a mapping 2

In the case {\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m} case, there is a linear map, which is “linear approximation” of {\psi}. In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If {M} and {N} are smooth manifolds and {\psi \colon M \rightarrow N} is a smooth map then for each {m \in M}, the map

\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N

is defined by

\displaystyle d\psi(v)(f) = v(f \circ \psi)

is called the pushforward. Actually,

\displaystyle d\psi \colon TM \rightarrow TN.

Suppose that {\dim{M} \ge \dim{N}} and {f \colon M \rightarrow N} is a differentiable mapping. We have

Definition 1 The mapping {f} is called a trivial fibration (differentiable) on {N} if there exists a differential manifold {F}, is called fibre of {f}, and a diffeomorphism

\displaystyle \phi \colon M \rightarrow N \times F

such that the following diagram is commutative

sodo

Một ví dụ về mặt chính quy

BT. Cho hàm f(x,y,z)=z^{2}. CMR 0  không là giá trị chính qui của hàm f thế nhưng f^{-1}(0) vẫn là mặt chính qui.

Lời giải. 

Ta có ma trận (f_x \ f_y \ f_z) = (0 \ 0 \ 2z) suy biến tức rankA < 1 khi và chỉ khi z=0, do đó tại điểm (x, y, 0), \forall x, y \in \mathbb{R}, ma trận trên suy biến và do đó là điểm kì dị. Mà f(x,y,0)=0 nên 0 là giá trị tới hạn, hay ko phải giá trị chính qui. Nhưng f^{-1}(0) là mặt phẳng Oxy, đây là mặt trơn nên chính qui.

The tangent plane of a surface and the tangent space of a manifold (at one point)

The tangent plane of a surface
By a tangent vector to S at a point p \in S, we mean the tangent vector \alpha'(0) of a differentiable parametrized curve \alpha: (-\epsilon, \epsilon) \to S with \alpha(0) = p.

What is the tangent plane of a surface? That is a plane which containes all of the tangent vectors of this surface at point p \in S.

Proposition
Let \varphi: U \subset \mathbb{R}^2 \to S be a parametrization of a regular surface S and let q \in U. The vector subspace of dim 2,

d\varphi_q(\mathbb{R}^2) \subset \mathbb{R}^3,

coincides with the set of tangent vectors to S at \varphi(q)

There is a similar situation in here. In the case of manifold, the tangent space is also built from the set of all of tangent vectors of a manifold.

Tangent spaces of a manifold

In Milnor’s book (\cite{Milnor}), the tangent space TM_x at x for arbitrary smooth manifold M \subset \mathbb{R}^k is defined:

Choose a parametrization g : U \to M \subset \mathbb{R}^k of a neighborhood g(U) of x in M, with g(u) = x. We have dg_u: \mathbb{R}^m \to \mathbb{R}^k. So the image dg_u(\mathbb{R}^m) of dg_u is equal to TM_x.

References

J. W. Milnor, Topology from differentiable viewpoint, 1965.

M. do Carmo, Differential geometry, curves and surfaces, 1976.