mathematics

Projective differential geometry

Projective differential geometry 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 Projective differential geometry rather than just read about it. In short: In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group. This is a mixture of the approaches from Riemannian geometry of studying invariances, and of the Erlangen program of characterizing geometries accordi…

Key takeaways

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

Reference excerpt

In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group. This is a mixture of the approaches from Riemannian geometry of studying invariances, and of the Erlangen program of characterizing geometries according to their group symmetries. The area was much studied by mathematicians from around 1890 for a generation (by J. G. Darboux, George Henri Halphen, Ernest Julius Wilczynski, E. Bompiani, G. Fubini, Eduard Čech, amongst others), without a comprehensive theory of differential invariants emerging. Élie Cartan formulated the idea of a general projective connection, as part of his method of moving frames; abstractly speaking, this is the level of generality at which the Erlangen program can be reconciled with differential geometry, while it also develops the oldest part of the theory (for the projective line), namely the Schwarzian derivative, the simplest projective differential invariant. Further work from the 1930s onwards was carried out by J. Kanitani, Shiing-Shen Chern, A. P. Norden, G. Bol, S. P. Finikov and G. F. Laptev. Even the basic results on osculation of curves, a manifestly projective-invariant topic, lack any comprehensive theory. The ideas of projective differential geometry recur in mathematics and its applications, but the formulations given are still rooted in the language of the early twentieth century.

Worked examples

Example 1 — a first encounter with Projective differential geometry

Start with the simplest possible case. Write down what Projective differential geometry 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 Projective differential geometry 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 Projective differential geometry 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 Projective differential geometry

In research: Projective differential geometry 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 Projective differential geometry 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: Projective differential geometry is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential geometry, Projective geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Projective differential geometry 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 Projective differential geometry in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Projective differential geometry 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 Projective differential geometry out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Projective differential geometry in simple terms?

Why does Projective differential geometry 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 Projective differential geometry?

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 Projective differential geometry.