Manifolds 19 | Tangent Space for Submanifolds [dark version]

TL;DR

For each point P on a k-dimensional submanifold M inside 9n, the tangent space is defined as the range of the differential of a local parameterization that maps 9k into 9n.

Briefing Cornell Notes

Briefing

Tangent spaces for submanifolds in 9n are built by taking the range of the differential of a local parameterization, turning the geometry of a curved set into a concrete linear subspace of 9n. For each point P on a k-dimensional submanifold M inside 9n, one chooses a local parameterization a6 from 9k into 9n that flattens M near P. The tangent space Ta6(P) is then defined as the image of the Jacobian (the differential) of a6 at the parameter value corresponding to P. Because the differential is a linear map from 9k to 9n, its range is a linear subspace of 9n with dimension k, giving a precise way to attach “directional” information to every point on the manifold.

The construction starts from the submanifold definition: locally, M can be “flattened” by a homeomorphism H that makes the manifold look like 9k inside 9n. Using H’s inverse, one restricts to the 9k slice to obtain a local parameterization a6. This parameterization is differentiable because it is built from smooth manifold charts, so derivatives and Jacobians behave exactly like in multivariable calculus. A simple mental picture is that the tangent space is the linear space you get by translating the best local linear approximation to the point P—analogous to how derivatives encode local linear behavior on curves.

A concrete example uses the circle S1 in 92. Parameterizing the circle by a6(t) = (cos t, sin t) produces a differentiable map from 9 to 92, and the tangent space at each point becomes the line in 92 spanned by the derivative direction at that parameter value. The key takeaway is that the tangent space is not abstract: it lives inside the ambient space 9n as an actual subspace.

The transcript then works through a higher-dimensional example: a surface in 93 given as the graph of a function f: 92 9to 9 (with at least C1 regularity). The submanifold M is the set of points (x, y, f(x, y)). A global parameterization a6(x, y) = (x, y, f(x, y)) has Jacobian matrix whose columns are the partial derivatives with respect to x and y. Those two column vectors—(1, 0, b1f/b1x) and (0, 1, b1f/b1y), evaluated at (x, y)—span the tangent space at the point (x, y, f(x, y)).

This tangent-space definition is presented as the bridge to abstract manifolds. If tangent spaces can be defined for submanifolds via differentials of smooth maps, then the same philosophy can extend beyond embedded sets: calculus on manifolds depends on having a consistent notion of derivatives and local linearization. Tangent spaces provide exactly that local linear approximation, which is slated for deeper discussion next.

Cornell Notes

For a k-dimensional submanifold M inside 9n, each point P gets a tangent space Ta6P that is a linear subspace of 9n. The method uses a local parameterization a6: 9k 9to 9n that flattens M near P, obtained from charts that make the manifold look like 9k locally. Let a6(P) = P, where P is the corresponding parameter value; then Ta6P is defined as the range of the differential da6 at P. Concretely, Ta6P is the span of the Jacobian matrix columns of a6 at P. This turns curvature into a computable linear object and sets up tangent spaces for abstract manifolds.

How does the tangent space Ta6P for a submanifold M 9n get defined from a parameterization?

Pick a local parameterization a6: 9k 9to 9n whose image describes M near P. Let P be the parameter value with a6(P) = P (possible because a6 is locally bijective). The tangent space is the range of the differential da6 at P, equivalently the image of the Jacobian matrix of a6 at P. Since da6 maps 9k linearly into 9n, its range is a k-dimensional linear subspace of 9n.

Why does “flattening” the manifold matter for constructing tangent spaces?

Flattening means choosing charts so that, locally, the submanifold looks like 9k inside 9n. A homeomorphism H can be used to make the manifold appear as 9k (with the remaining coordinates set to zero). Using H’s inverse and restricting to that 9k slice produces a local parameterization a6 into 9n. That parameterization is what the differential is taken of, so flattening is the geometric step that makes the tangent space construction concrete.

What does the tangent space look like for the circle S1 in 92?

Using the standard parameterization a6(t) = (cos t, sin t), the map goes from 9 to 92. The differential da6(t) is a linear map from 9 to 92, so its range is a 1-dimensional subspace (a line) in 92. Geometrically, it is the direction of the circle at the corresponding point, translated to that point.

How is the tangent space computed for a surface in 93 given by a graph z = f(x, y)?

Let M be the graph of f: 92 9to 9 with at least C1 regularity, so M = {(x, y, f(x, y))}. Use the global parameterization a6(x, y) = (x, y, f(x, y)). The Jacobian matrix has two columns: the partial derivative with respect to x gives (1, 0, b1f/b1x) and the partial derivative with respect to y gives (0, 1, b1f/b1y), both evaluated at (x, y). The tangent space at P = (x, y, f(x, y)) is the span of these two vectors.

What is the conceptual payoff of tangent spaces for abstract manifolds?

The construction for submanifolds shows how to attach linear spaces to points using differentials of smooth maps. That same mechanism is what makes calculus on manifolds possible: derivatives require a linear structure, and tangent spaces provide a local linearization of the curved object. Once this works for embedded submanifolds, the goal is to define an analogous tangent-space concept for abstract manifolds so that smooth maps and differentials still make sense.

Review Questions

Given a local parameterization a6: 9k 9to 9n with a6(P)=P, what exact linear-algebra operation defines Ta6P?
For the graph surface (x, y, f(x, y)) in 93, which two Jacobian column vectors span the tangent space at (x, y, f(x, y))?
Why does the tangent space end up as a subspace of 9n rather than an abstract object?

Key Points

1
For each point P on a k-dimensional submanifold M inside 9n, the tangent space is defined as the range of the differential of a local parameterization that maps 9k into 9n.
2
Local parameterizations come from flattening the manifold so it looks like 9k inside 9n near P, then using the inverse of the flattening map to build a6.
3
The tangent space Ta6P is a linear subspace of 9n with dimension k because the differential is a linear map from 9k to 9n.
4
For the circle S1 in 92, the tangent space at a point is the line in 92 spanned by the derivative direction of a6(t) = (cos t, sin t).
5
For a surface in 93 given by z = f(x, y), the tangent space at (x, y, f(x, y)) is spanned by (1, 0, b1f/b1x) and (0, 1, b1f/b1y).
6
Tangent spaces provide a local linear approximation of a manifold, enabling calculus on curved sets.
7
The submanifold construction is positioned as the stepping stone toward defining tangent spaces for abstract manifolds.

Highlights

Tangent spaces for submanifolds are defined as the image of the Jacobian (differential) of a local parameterization at the parameter value corresponding to the point.

Flattening a submanifold locally to look like 9k inside 9n is what produces the parameterization needed for the tangent-space definition.

For a graph surface (x, y, f(x, y)) in 93, the tangent space is spanned directly by the two partial-derivative vectors from the Jacobian columns.

The tangent space lives inside the ambient space 9n, making it a concrete linear object rather than an abstract construction.