Calculus Definitions >

**Contents:**

## What is a Manifold?

In beginning mathematics classes, you learn about two-dimensional surfaces and curves (like arcs or circles) or three-dimensional objects (like cones or spheres). In higher dimensional space, these curves and surfaces are called **manifolds**.

## Simple Example of a Manifold in Mathematics

The definition of a manifold “…involves rather a large number of technical details” (Lee, 2006). But it’s possible to grasp the idea of a manifold without trudging through the depths of a technical definition, because the simplest higher-dimensional manifold in mathematics actually happens in three-dimensional space.

For example, the following image shows a single **affine motion**, a translation and rotation of a cone:

If you take the set of **all possible affine motions **in three-dimensional space, you get a **six-dimensional manifold** (Tu, 2010).

## Dimension of a Manifold

The *dimension* of a manifold in mathematics is the number of parameters (i.e. independent numbers) needed to plot a point in space.

A line is a simple **manifold of dimension 1**. To plot the number 2 on a number line only requires one number: 2. Although a line isn’t “curved” in the usual sense of the world, it’s still considered a curve in the realm of manifolds. A graph of a continuous function also is a manifold of dimension 1.

Planes, spheres, cylinders and similar objects are **manifolds of dimension 2. **These manifolds are *not* the solid shapes but rather the skin that encloses the shape (like the surface of the Earth). For these shapes, two points (i.e. Cartesian coordinates) are needed to specify a point.

Any dimension above 3 is difficult or impossible to visualize. However, they still adhere to the same rule for the number of points. For example, a **7-dimensional manifold**, modeled locally on ℝ^{7} (the doublestruck R is the set of all reals), would need 7 numbers to specify its location as long as it isn’t too far away from the starting point.

## Covariant Derivative

A **covariant derivative** (∇_{x)} generalizes an ordinary derivative (i.e. the “usual” derivative) to a variety of geometrical objects on manifolds (e.g. forms, tensors, or vectors).

More specifically, the covariant derivative is an operator that reduces to a flat space partial derivative with Cartesian coordinates; It transforms as a tensor on a random manifold and in fact, **the essential feature of a covariant derivative is this tensor characteristic** (Wempner, 1982). While the “usual” types of functions and vector fields are dealt with by ordinary differentiation, covariant derivatives specifically deal with differentiating tensors.

## Covariant Derivative Defined More Simply

In (relatively) simple terms, the covariant derivative tells you how the “head” of a vector moves in relation to some movement of the vector’s “tail” in curved spacetime. A covariant derivative allows you to construct directional derivatives by just contracting with a vector. You don’t need to know its derivatives.

The covariant derivative is a rule that takes as inputs:

- A vector, defined at point P,
- A vector field, defined in the neighborhood of P.

The output is also a vector at point P.

**Terminology note**: In (relatively) simple terms, a **tensor** is very similar to a vector, with an array of components that are functions of a space’s coordinates. A **manifold** is a non-Euclidean space that, close up, looks Euclidean (i.e. flat).

## References

Lee, 2006. Introduction to Topological Methods. Springer Science and Business Media.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 48-50, 1953.

Schmutzer, E. Relativistische Physik (Klassische Theorie). Leipzig, Germany: Akademische Verlagsgesellschaft, 1968.

Tu, L. (2010). An Introduction to Manifolds. Springer Science & Business Media.

Weinberg, S. “Covariant Differentiation.” §4.6 in Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, pp. 103-106, 1972.

Wempner, G. (1982). Mechanics of Solids with Applications to Thin Bodies. Springer Science & Business Media.

GR lecture 5 Covariant derivatives, Christoffel connection, geodesics, electromagnetism in curved spacetime, local conservation of 4-momentum