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Chapter 2: Problem 8

Prove that if \(L: R^{3} \rightarrow R^{3}\) is a linear map and \(S \subset R^{3}\) is a regular surface invariant under \(L\), i.e., \(L(S) \subset S\), then the restriction \(L \mid S\) is a differentiable map and $$ d L_{p}(w)=L(w), \quad p \in S, w \in T_{p}(S) . $$

Short Answer

Expert verified
The restriction \( L|_S \) is differentiable, and \( dL_p(w) = L(w) \) for all \( p \in S \) and \( w \in T_p(S) \).

Step by step solution

01

Understand the given information

We are given a linear map \( L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \) and a regular surface \( S \subset \mathbb{R}^{3} \) such that \( L(S) \subset S \). We need to prove that the restriction of \( L \) to \( S \), denoted as \( L|_S \), is a differentiable map and satisfies the condition \( dL_p(w) = L(w) \) for all \( p \in S \) and \( w \in T_p(S) \).
02

Definition of a regular surface

Recall that a regular surface is a subset \( S \subset \mathbb{R}^{3} \) such that for each point \( p \in S \), there exists a neighborhood \( U \subset \mathbb{R}^{3} \) and a differentiable function \( \phi: U \rightarrow \mathbb{R} \) with non-zero gradient such that \( S \cap U = \{ x \in U : \phi(x) = 0 \} \).
03

Restriction of a linear map

Since \( L \) is a linear map from \( \mathbb{R}^{3} \) to \( \mathbb{R}^{3} \), it is differentiable everywhere. The restriction \( L|_S \) refers to considering the map \( L \) only on points lying on the surface \( S \).
04

Differentiability of the restriction

Since \( S \) is invariant under \( L \), \( L(S) \subset S \), the image of any point \( p \in S \) under \( L \) is also a point in \( S \). Thus, the restriction map \( L|_S : S \rightarrow S \) remains differentiable because it is simply the original differentiable map considered over a subset where it is well-defined.
05

Tangent vectors and linearity

For each point \( p \in S \), the tangent space at \( p \), denoted as \( T_p(S) \), is a subspace of \( \mathbb{R}^{3} \). Since \( L \) is linear, for any tangent vector \( w \in T_p(S) \), \( L(w) \) will still satisfy the linearity condition, thus \( L \) maps tangent vectors to tangent vectors.
06

Conclusion with differential

Because \( L \) is a linear map and differentiable, the differential \( dL_p \) at point \( p \) is simply \( L \). Therefore, for any \( w \in T_p(S) \), the derivative \( dL_p(w) = L(w) \). This means \( L \) not only maps points on the surface to points on the surface but also maps tangent vectors correctly, maintaining differentiability and linearity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear maps
A linear map, or linear transformation, is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In simpler terms, applying a linear map to a combination of vectors results in the same combination of their images. Suppose we have a linear map denoted by \( L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \), then for any two vectors \( u \) and \( v \) from \( \mathbb{R}^{3} \), and any scalar \( c \), the map satisfies:
  • \( L(u + v) = L(u) + L(v) \)
  • \( L(cu) = cL(u) \)
In the context of our exercise, the linear map \( L \) helps us by ensuring that the map is differentiable everywhere in \( \mathbb{R}^{3} \), which is crucial for our proofs involving differentiability on more restricted sets like regular surfaces.
regular surfaces
A regular surface is a subset of \( \mathbb{R}^{3} \) that locally resembles a two-dimensional plane. To be more precise, for each point \( p \) on a regular surface \( S \), there exists a neighborhood \( U \) in \( \mathbb{R}^{3} \) and a differentiable function \( \phi: U \rightarrow \mathbb{R} \) such that:
  • The gradient of \( \phi \) is non-zero: \( abla \phi eq 0 \)
  • The surface can be expressed as \( S \cap U = \{ x \in U : \phi(x) = 0 \} \)
This means that around each point, we can describe the surface using the equation \( \phi(x) = 0 \), and the condition of the non-zero gradient ensures that the surface is smooth without any sharp corners or cusps.
tangent spaces
Tangent spaces are a crucial concept in differential geometry, capturing the idea of a plane that 'touches' a surface at a single point and extends in the directions tangent to the surface. For a regular surface \( S \) in \( \mathbb{R}^{3} \), the tangent space at a point \( p \), denoted by \( T_p(S) \), is the set of all vectors that are tangent to the surface at that point.
Formally, if \( p \in S \), a tangent vector \( w \) lies in \( T_p(S) \) if it satisfies all of the differential constraints posed by the surface at that point. Using the surface defined by \( \phi(x) = 0 \), any tangent vector \( w \) at \( p \) must satisfy \( abla \phi(p) \cdot w = 0 \). Given the linearity of \( L \), if \( w \in T_p(S) \), then \( L(w) \) will map back into the tangent space, ensuring consistent application of the linear map on the surface's tangent vectors.
differentiability
Differentiability refers to the existence of a well-defined tangent plane or space at every point of a set, allowing the application of calculus techniques. A map is differentiable if it has a linear approximation near every point. For our current discussion, the linear map \( L \) is differentiable everywhere by definition, which is significant when considering its restriction to a subset like a regular surface.
When we discuss the differentiability of the restriction of \( L \) to the surface \( S \), \( L|_S \), we mean that at each point \( p \) on \( S \), the map behaves smoothly, and the differential, denoted by \( dL_p \), exists and equals \( L \). This ensures that the behavior of the map does not 'break' at any point on the surface and that it maps tangent vectors in a manner preserving their directional properties.

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Most popular questions from this chapter

Let \(w\) be a tangent vector to a regular surface \(S\) at a point \(p \in S\) and let \(\mathbf{x}(u, v)\) and \(\overline{\mathbf{x}}(\bar{u}, \bar{v})\) be two parametrizatioos at \(p\). Suppose that the expressions of \(w\) in the bases associated to \(\mathbf{x}(u, v)\) and \(\overline{\mathbf{x}}(\bar{u}, \bar{v})\) are $$ w=\alpha_{1} \mathbf{x}_{u}+\alpha_{2} \mathbf{x}_{v} $$ and $$ w=\beta_{1} \overline{\mathbf{x}}_{\bar{u}}+\beta_{2} \mathbf{x}_{\bar{v}} $$ Show that the coordinates of \(w\) are related by $$ \begin{aligned} \beta_{1} &=\alpha_{1} \frac{\partial \bar{u}}{\partial u}+\alpha_{2} \frac{\partial \bar{u}}{\partial v} \\ \beta_{2} &=\alpha_{1} \frac{\partial \bar{v}}{\partial u}+\alpha_{2} \frac{\partial \bar{v}}{\partial v} \end{aligned} $$ where \(\bar{u}=\bar{u}(u, v)\) and \(\bar{v}=\bar{v}(u, v)\) are the expressions of the change of coordinates.

Let \(C\) be a regular curve and let \(\alpha: I \subset R \rightarrow C, \beta: J \subset R \rightarrow C\) be two parametrizations of \(C\) in a neighborhood of \(p \in \alpha(I) \cap \beta(J)=W\). Let $$ h=\alpha^{-1} \circ \beta: \beta^{-l}(W) \rightarrow \alpha^{-l}(W) $$ be the change of parameters. Prove that a. \(h\) is a diffeomorphism. b. The absolute value of the arc length of \(C\) in \(W\) does not depend on which parametrization is chosen to define it, that is, $$ \left|\int_{t_{0}}^{t}\right| \alpha^{\prime}(t)|d t|=\left|\int_{\tau_{0}}^{\tau}\right| \beta^{\prime}(\tau)|d \tau|, \quad t=h(\tau), t \in I, \tau \in J . $$

Let \(S^{2}=\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{2}+z^{2}=1\right\\}\) be the unit sphere and let \(A: S^{2} \rightarrow S^{2}\) be the (antipodal) map \(A(x, y, z)=(-x,-y,-z)\). Prove that \(A\) is a diffeomorphism.

Let \(S\) be a regular surface covered by coordinate neighborhoods \(V_{1}\) and \(V_{2}\). Assume that \(V_{1} \cap V_{2}\) has two connected components, \(W_{1}, W_{2}\), and that the Jacobian of the change of coordinates is positive in \(W_{1}\) and negative in \(W_{2}\). Prove that \(S\) is nonorientable.

Show that the paraboloid \(z=x^{2}+y^{2}\) is diffeomorphic to a plane.
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