Tangent Space Of A Hypersurface Defined By Equation

In the realm of differential geometry and analysis, hypersurfaces hold a position of significant importance. They represent a natural extension of surfaces into higher dimensions, and understanding their properties is crucial for various applications in physics, engineering, and computer graphics. One fundamental concept associated with hypersurfaces is the tangent space, which provides a local linear approximation of the hypersurface at a given point. This article delves into the intricacies of tangent spaces, particularly focusing on hypersurfaces defined by equations, and aims to provide a comprehensive understanding of their structure and properties.

Defining Hypersurfaces and Submersions

Before embarking on the exploration of tangent spaces, it's essential to lay a solid foundation by defining the key concepts involved. A hypersurface is an $(n-1)$-dimensional manifold embedded in an $n$-dimensional space. In simpler terms, it's a surface-like object that exists in a higher-dimensional space. For instance, a sphere in three-dimensional space is a hypersurface, as it's a two-dimensional surface embedded in a three-dimensional space. Similarly, a plane in three-dimensional space is also a hypersurface.

To formally define a hypersurface using equations, we introduce the concept of a submersion. Let $G: U o ext{ℝ}^m$ be a smooth map, where $U$ is an open subset of $ ext{ℝ}^n$. The map $G$ is said to be a submersion if its differential, denoted by $dG_p$, is surjective for all points $p$ in $U$. In simpler terms, the differential $dG_p$ maps the tangent space at $p$ onto the entire target space $ ext{ℝ}^m$. The condition of surjectivity ensures that the level sets of $G$ are manifolds.

A crucial result in differential geometry states that if $G: U o ext{ℝ}^m$ is a submersion, then the set $M = \{p ext{∈} U : G(p) = 0 \}$, which represents the set of points where $G$ vanishes, is a manifold of dimension $n - m$. This manifold $M$ is precisely the hypersurface we are interested in. The equation $G(p) = 0$ implicitly defines the hypersurface, and the submersion condition guarantees that the resulting set is indeed a manifold.

For example, consider the map $G: ext{ℝ}^3 o ext{ℝ}$ defined by $G(x, y, z) = x^2 + y^2 + z^2 - 1$. The set $M = \{(x, y, z) ext{∈} ext{ℝ}^3 : G(x, y, z) = 0\}$ represents the unit sphere in three-dimensional space. It can be shown that $G$ is a submersion, and thus the unit sphere is a manifold of dimension 2 (which is $3 - 1$). This example illustrates how the concept of submersions allows us to define hypersurfaces using equations.

Delving into Tangent Spaces

Now that we have a clear understanding of hypersurfaces defined by equations, we can delve into the core concept of tangent spaces. The tangent space at a point on a manifold is a vector space that captures the local linear structure of the manifold at that point. Intuitively, the tangent space can be thought of as the set of all possible directions one can move along the manifold starting from that point. It provides a linear approximation of the manifold in a neighborhood of the point.

Formally, let $M$ be a manifold embedded in $ ext{ℝ}^n$, and let $p$ be a point on $M$. The tangent space of $M$ at $p$, denoted by $T_pM$, is defined as the set of all tangent vectors to smooth curves on $M$ that pass through $p$. A tangent vector is simply the velocity vector of a smooth curve at a particular point. In other words, if $\gamma: (-\epsilon, \epsilon) o M$ is a smooth curve on $M$ such that $\gamma(0) = p$, then the velocity vector $\gamma'(0)$ is a tangent vector to $M$ at $p$. The collection of all such tangent vectors forms the tangent space $T_pM$.

Tangent spaces are vector spaces, meaning that they satisfy the axioms of vector spaces, such as closure under addition and scalar multiplication. This vector space structure allows us to perform linear operations on tangent vectors, which is crucial for many applications in differential geometry and related fields. The dimension of the tangent space $T_pM$ is equal to the dimension of the manifold $M$.

For instance, consider the unit sphere in three-dimensional space. At any point $p$ on the sphere, the tangent space $T_pM$ is a two-dimensional plane that is tangent to the sphere at $p$. This plane represents the set of all possible directions one can move along the sphere starting from the point $p$. The tangent space provides a local linear approximation of the sphere at that point.

Tangent Space of a Hypersurface Defined by an Equation

Now, let's focus on the specific case of a hypersurface $M$ defined by an equation, as described earlier. Suppose $M = \{p ext{∈} U : G(p) = 0\}$, where $G: U o ext{ℝ}^m$ is a submersion. Our goal is to understand the structure of the tangent space $T_pM$ for this particular type of manifold. The key insight lies in the relationship between the differential of $G$ and the tangent space.

Let $\gamma: (-\epsilon, \epsilon) o M$ be a smooth curve on $M$ such that $\gamma(0) = p$. Since $\gamma$ lies on $M$, we have $G(\gamma(t)) = 0$ for all $t$ in $(-\epsilon, \epsilon)$. This means that the composition of $G$ and $\gamma$ is a constant function, specifically the zero function. Differentiating both sides of this equation with respect to $t$ using the chain rule, we obtain:

$dG_{\gamma(t)}(\gamma'(t)) = 0$

Evaluating this equation at $t = 0$, we get:

$dG_p(\gamma'(0)) = 0$

This equation reveals a crucial relationship: the differential of $G$ at $p$, applied to the tangent vector $\gamma'(0)$, yields the zero vector. In other words, the tangent vector $\gamma'(0)$ lies in the kernel (null space) of the linear map $dG_p$. This observation leads us to a fundamental characterization of the tangent space $T_pM$.

Specifically, the tangent space $T_pM$ is precisely the kernel of the differential $dG_p$. That is:

$T_pM = ext{ker}(dG_p) = \{v ext{∈} ext{ℝ}^n : dG_p(v) = 0\}$

This result provides a powerful tool for computing and understanding the tangent space of a hypersurface defined by an equation. It tells us that the tangent space consists of all vectors in $ ext{ℝ}^n$ that are mapped to the zero vector by the differential $dG_p$. In geometric terms, these are the vectors that are