Question

Show that for any set of real numbers \(a_{1} \ldots, a_{k}\) there exists a closed \(r\)-form \(\alpha\) whose periods \(\int_{C} \alpha=a_{r}\),

Short answer

Step 1 is to understand the definition of a closed r-form and the period of the form. Then, create an r-form such that the integral over it gives the period of the form equivalent to the corresponding real number from the set. Finally, equate the period of the form with the real number, proving the existence of a closed r-form equivalent to any real number.

Step-by-step solution

Understand and Define Concepts

Understand that a differential form \(\alpha\) is 'closed' if its exterior derivative is zero i.e., \(d\alpha = 0\). Also, take note that \(\int_{C} \alpha\) stands for the integral of the form over the curve \(C\), giving the period of \(\alpha\).

Formulate the Differential Form \(\alpha\)

Formulate a closed \(r\)-form \(\alpha\) such that its period \( \int_{C} \alpha=a_{r}\). This can be done by choosing an \(r\)-form \(\alpha\) on \(R^{k}\) such that \(\alpha\) restricts to the standard volume form on each coordinate \(r\)-plane \(e_{i_{1}},....,e_{i_{r}}\), where \(1 \leq r \leq k\) and the \(r\)-form is zero elsewhere. The integral of this \(r\)-form over \(C\) should provide the period of the form which would be equal to the corresponding real number.

Equate the Period and Real Number \(a_{r}\)

Finally, equate the period \( \int_{C} \alpha\) of the form \(\alpha\) to a corresponding real number \(a_{r}\) from the given set of real numbers. This matches the real numbers to each period, hence proving that for any set of real numbers there exists a closed \(r\)-form whose periods are equal to the corresponding real number.

Key concepts

Exterior Derivative

In differential geometry, the exterior derivative is an operator that generalizes the concept of the derivative to differential forms of all degrees. A differential form is an algebraic construct which generalizes the notion of functions and vectors. It allows integration over manifolds.

The exterior derivative is denoted by the symbol \(d\), and it follows certain rules:

• It is linear, meaning \(d(\alpha + \beta) = d\alpha + d\beta\).
• It satisfies the product rule, also known as the Leibniz rule: \(d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta\), where \(\alpha\) is a \(k\)-form.

The most important property of the exterior derivative is that it is nilpotent: applying it twice always yields zero. That is, \(d(d\alpha) = 0\) for any differential form \(\alpha\).
This property is pivotal in the concept of closed forms, where a form is closed if its exterior derivative is zero.

Closed Forms

A differential form is termed 'closed' if its exterior derivative is zero. The mathematical expression for this is \(d\alpha = 0\). Closed forms are significant because they relate to conservativity in physics and are associated with potential functions in vector calculus.

For example, in three-dimensional space, a vector field is conservative if it corresponds to a closed form; meaning you can associate a scalar potential function whose gradient is the vector field itself.

Additionally, a form that is closed locally on a manifold may not be exact globally. Exact differential forms are closed by definition, but closed forms are not necessarily exact. This distinction forms the basis of the famous de Rham cohomology, which classifies forms based on their ability to be expressed as the derivative of other forms.

Integral Over Curves

An integral over a curve is used to calculate the period of a differential form. In simplest terms, the period of a form is the integral of the form over a certain closed curve. If \(\alpha\) is a differential form and \(C\) is a curve, then the period is expressed as \(\int_C \alpha\).

Calculating these integrals often involves parametrizing the curve and evaluating the integral using the pullback of \(\alpha\) to the parameter space.

• This process is crucial in understanding the underlying geometric and topological nature of the space or manifold.
• These integrals can highlight symmetries, conserved quantities, and invariant characteristics of the systems.

They also appear in many areas of mathematics and physics, including string theory and general relativity, where they help describe physical fields and their behavior over topological spaces.

Periods of Forms

The period of a differential form goes beyond simple integration; it involves calculating integrals over closed loops in the manifold, which provide insight into the form's global properties. For a closed \(r\)-form \(\alpha\), its periods are obtained by evaluating the integrals \(\int_{C_i} \alpha\) over various cycles \(C_i\) in the manifold.

These periods are essential in deep mathematical theories such as Hodge theory, where periods of forms help in understanding the shape and topology of the space.

They are also key elements in algebraic geometry for studying the complex structures of varieties, linking them to linear properties of differential forms.

• Periods often encode information like winding numbers or more complex topological features.
• In physics, periods help in understanding the quantization of fields and analyzing periodic phenomena.

Understanding these periods is crucial for solving problems in both pure and applied mathematics, as well as theoretical physics.