Chapter 17: Problem 11

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

Expert verified

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.