mathematics

Wiedersehen pair

Wiedersehen pair is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Wiedersehen pair rather than just read about it. In short: In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such that every geodesic through x also passes through y, and the same with x and y interchanged. For example, on an ordinary sphere where the geodesics are great circles, the Wiedersehen pairs are exactly the pairs of anti…

Key takeaways

Wiedersehen pair belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Wiedersehen pair to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Wiedersehen pair from memory before moving on to harder problems.

Reference excerpt

In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such that every geodesic through x also passes through y, and the same with x and y interchanged. For example, on an ordinary sphere where the geodesics are great circles, the Wiedersehen pairs are exactly the pairs of antipodal points. If every point of an oriented manifold (M, g) belongs to a Wiedersehen pair, then (M, g) is said to be a Wiedersehen manifold. The concept was introduced by the Austro-Hungarian mathematician Wilhelm Blaschke and comes from the German term meaning "seeing again". As it turns out, in each dimension n the only Wiedersehen manifold (up to isometry) is the standard Euclidean n-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of Berger, Kazdan, Weinstein (for even n), and Yang (odd n).

See also Cut locus (Riemannian manifold)

References Berger, Marcel (1978). "Blaschke's Conjecture for Sphere". Manifolds all of whose Geodesics are Closed. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 236–242. doi:10.1007/978-3-642-61876-5_13. ISBN 978-3-642-61878-9. Blaschke, Wilhelm (1921). Vorlesung über Differentialgeometrie I. Berlin: Springer-Verlag. Kazdan, Jerry L. (1982). "An isoperimetric inequality and Wiedersehen manifolds". Seminar on Differential Geometry. (AM-102). Princeton University Press. pp. 143–158. ISBN 978-0-691-08268-4. JSTOR j.ctt1bd6kkq.9. Retrieved 2024-01-29. McKay, Benjamin. "Summary of progress on the Blaschke conjecture" (PDF). Retrieved 2024-01-29. Weinstein, Alan (1974-01-01). "On the volume of manifolds all of whose geodesics are closed". Journal of Differential Geometry. 9 (4). doi:10.4310/jdg/1214432547. ISSN 0022-040X. C. T. Yang (1980). "Odd-dimensional wiedersehen manifolds are spheres". J. Differential Geom. 15 (1): 91–96. doi:10.4310/jdg/1214435386. ISSN 0022-040X. Chavel, Isaac (2006). Riemannian geometry: a modern introduction. New York: Cambridge University Press. pp. 328–329. ISBN 0-521-61954-8.

External links Weisstein, Eric W. "Wiedersehen pair". MathWorld. Weisstein, Eric W. "Wiedersehen surface". MathWorld.

Worked examples

Example 1 — a first encounter with Wiedersehen pair

Start with the simplest possible case. Write down what Wiedersehen pair claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Wiedersehen pair before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Wiedersehen pair ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Wiedersehen pair

In research: Wiedersehen pair appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Wiedersehen pair in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Wiedersehen pair is common in secondary-school and first-year university syllabi. It links to neighbouring topics Equations, Manifolds, Riemannian geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Wiedersehen pair outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Wiedersehen pair in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Wiedersehen pair means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Wiedersehen pair out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Wiedersehen pair in simple terms?

Why does Wiedersehen pair matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Wiedersehen pair?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Wiedersehen pair.