Problem 15 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Is the map \(\alpha: \mathbb{R} \rightarrow \mathbb{R}^{2}\) given by \(x=\sin t, y=\sin 2 t\) (1) an immersion, (ii) an cmbedded submanifold?

Step-by-Step Solution

Answer

The map \( \alpha: \mathbb{R} \rightarrow \mathbb{R}^{2}\) given by \(x=\sin t,\)\(y=\sin 2 t\) is neither an immersion nor an embedded submanifold.

Step 1: Calculate the Jacobian matrix

The derivative of \( \alpha(t) \) is given by the Jacobian matrix of the component functions. So,we must take the derivatives of \(x=\sin t\) and \(y=\sin 2 t\), obtaining \(\begin{bmatrix} x'\ y' \end{bmatrix} = \begin{bmatrix} \cos t \ 2\cos 2t \end{bmatrix}\).

Step 2: Determine if the map is an immersion

For the map to be an immersion, the derivative, or Jacobian, must be injective. In this case, the Jacobian would be injective if the determinant of Jacobian matrix is not zero everywhere. However, it should be noted that the determinant of a matrix and whether a mapping is injective are related for square matrices only. And since in this case the Jacobian is a 1x2 matrix, it can't be injective, hence the map \( \alpha(t) \) is not an immersion.

Step 3: Check if the map is an embedded submanifold

For a map to be an embedded submanifold, it must first be an immersion, which as we determined is not the case. Therefore, the map \( \alpha(t) \) is not an embedded submanifold.

Key Concepts

Immersion

In the context of differential geometry, an immersion is a type of smooth map between manifolds that is locally injective. Think of it as a way of mapping shapes from one space to another without any overlaps.

To determine if our map \( \alpha(t) = (\sin t, \sin 2t) \) is an immersion, we calculate its derivative, known as the Jacobian matrix. This map is from \( \mathbb{R} \rightarrow \mathbb{R}^{2} \), so the Jacobian is usually a 1x2 matrix.

• If the Jacobian matrix has full rank (meaning it can be injective), the map is considered an immersion.

However, in our example, the Jacobian matrix \( \begin{bmatrix} \cos t & 2\cos 2t \end{bmatrix} \) doesn't meet the criteria. Hence, \( \alpha(t) \) is not an immersion.

Submanifold

A submanifold is essentially a manifold that is "contained" within another manifold of higher dimensions. For a map to describe an embedded submanifold, it needs to have certain properties.

An important requirement for a map like \( \alpha: \mathbb{R} \rightarrow \mathbb{R}^{2} \) to describe an embedded submanifold is first being an immersion. This is crucial because being an immersion ensures that around each point, the map has a unique 'direction' in the larger space it exists in.

• In this particular example, since \( \alpha(t) \) wasn't an immersion, it can't be an embedded submanifold.

Embedded submanifolds visually give you the sense of a surface or curve sitting perfectly inside a larger space without intersection.

Jacobian matrix

The Jacobian matrix is a crucial concept in differential geometry. It is used to express the derivative of a vector-valued function. In simpler terms, it is a matrix that contains all first-order partial derivatives of a vector function.

For a function \( \alpha(t) = (\sin t, \sin 2t) \), the Jacobian matrix is a 1x2 matrix. This matrix reflects how the components of the function change as the input variables change.

• The specific Jacobian for \( \alpha \) here is \( \begin{bmatrix} \cos t & 2\cos 2t \end{bmatrix} \).
• The properties of this matrix help determine if the function is an immersion or if it describes an embedded submanifold.

Understanding the Jacobian can help answer complex questions about the geometry of maps like these.