Question

Show that the map \(\alpha: \hat{R}^{2} \rightarrow \mathbb{R}^{3}\) defined by $$ u=x^{2}+y^{2}, \quad v=2 x y, \quad w=x^{2}-y^{2} $$ is an immersion Is it an embedded submanifold?

Short answer

The map \(\alpha\) is an immersion but not an embedded submanifold.

Step-by-step solution

Compute the Jacobian matrix

Begin by computing the Jacobian matrix of the map \(\alpha\), which is the matrix of the first order partial derivatives of \(\alpha\). The Jacobian matrix is thus given by \[J_{\alpha} =\begin{bmatrix}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y}\end{bmatrix} = \begin{bmatrix}2x & 2y \2y & 2x \2x & -2y\end{bmatrix}, \]provided that \(x\) and \(y\) are not both zero, since in that case the derivative is not defined.

Check the rank of the Jacobian matrix

We then need to check that the Jacobian matrix \(J_{\alpha}\) has full rank for all \((x,y) \in \mathbb{R}^2\). We can achieve this by computing the determinant of the Jacobian matrix \(J_{\alpha}\), given by\[\text{det}(J_{\alpha}) = 2x \cdot 2x - 2y \cdot 2y = 4x^{2} - 4y^{2} = 4(x^{2} - y^{2}).\]We get that \(\text{det}(J_{\alpha}) \neq 0\), hence the Jacobian matrix has full rank \(2\) everywhere except at the origin, and \(\alpha\) is an immersion.

Check if the map is an embedded submanifold

To check if the map \(\alpha\) represents an embedded submanifold, we need to ensure that the image of the map is a topological submanifold and the differential of the map is a homeomorphism. The latter is guaranteed by the fact that \(\alpha\) is an immersion. However, the former condition is, in general, harder to verify. In our case, the map \(\alpha\) is not a homeomorphism on its image, due to the zero Jacobian at the origin, thus it does not represent an embedded submanifold.

Key concepts

Immersion

The concept of immersion in differential geometry is about how one manifold can be smoothly mapped into another manifold. Think of it as placing a lower-dimensional object into a higher-dimensional space without crumpling it.- An immersion essentially ensures that locally, the map behaves like an embedding. - It implies the derivative (or Jacobian matrix of the map) has full rank everywhere. This means the map stretches but does not tear.- In our exercise, the map was from \(\hat{R}^{2} \to \mathbb{R}^{3}\). We verified that the Jacobian matrix has full rank almost everywhere, showing the map is an immersion except at the origin.Understanding immersion helps us comprehend how differently one structure can align with another in a higher-dimensional space.

Submanifold

A submanifold is a manifold that is a subset of another manifold, which also inherits certain properties from the larger manifold. Just as a plane or curve can sit inside three-dimensional space, a submanifold sits smoothly within a larger manifold.- For a map like in our exercise, submanifolds are important in understanding the geometric image created in the target space.- To be more specific, an embedded submanifold has to satisfy the condition of being homeomorphic to its image.- The map in our exercise, despite being an immersion, does not form an embedded submanifold of \( \mathbb{R}^{3} \).This is because while immersions make sure the map is smooth, they do not always ensure the image sits nicely without singularities or overlaps in its environment.

Jacobian Matrix

The Jacobian matrix is a key tool for understanding changes of variables in multi-variable functions. It holds the first partial derivatives of a function and helps us figure out how the function transforms regions from its domain to its range.- In our exercise, the Jacobian matrix was \[J_{\alpha} = \begin{bmatrix}2x & 2y \ 2y & 2x \ 2x & -2y\end{bmatrix}\].- Checking for full rank by ensuring the determinant is non-zero confirms that the transformation is locally invertible (at most points).- A full-rank condition is crucial for the function to be an immersion.- Practical aspects include understanding volume transformations and orientations in higher-dimensional spaces.The Jacobian matrix is like a tool that maps how small changes evolve under transformations.

Homeomorphism

Homeomorphism touches on the importance of topological equivalence in differential geometry. When two spaces are homeomorphic, they can be transformed into each other via continuous, bijective, and invertible functions. - In simpler terms, it tells us if two spaces are topologically the same, even if they look different geometrically. - For our function, a homeomorphism would suggest that there are no holes or overlaps in the image manifold. - The map in the exercise, however, doesn't sustain a homeomorphism due to issues at the origin where the Jacobian fails to maintain its rank. Understanding homeomorphism is crucial for determining not just the likeness of shapes, but their inherent structural compatibility. This also influences the property of being an embedded submanifold in its target space.