mathematics

Hilbert's theorem (differential geometry)

Hilbert's theorem (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 Hilbert's theorem (differential geometry) rather than just read about it. In short: In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S {\displaystyle S} of constant negative gaussian curvature K {\displaystyle K} immersed in R 3 {\displaystyle \mathbb {R} ^{3}} . This theorem answers the question for the negative case of which surfaces in R 3 {\displaystyle \mathbb {R} ^{3}} can be obtained by isometrically immersing complete manifolds with con…

Key takeaways

Hilbert's theorem (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 Hilbert's theorem (differential geometry) to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hilbert's theorem (differential geometry) from memory before moving on to harder problems.

Reference excerpt

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S {\displaystyle S} of constant negative gaussian curvature K {\displaystyle K} immersed in R 3 {\displaystyle \mathbb {R} ^{3}} . This theorem answers the question for the negative case of which surfaces in R 3 {\displaystyle \mathbb {R} ^{3}} can be obtained by isometrically immersing complete manifolds with constant curvature.

History Hilbert's theorem was first treated by David Hilbert in "Über Flächen von konstanter Krümmung" (Trans. Amer. Math. Soc. 2 (1901), 87–99). A different proof was given shortly after by E. Holmgren in "Sur les surfaces à courbure constante négative" (1902). A far-leading generalization was obtained by Nikolai Efimov in 1975.

Proof The proof of Hilbert's theorem is elaborate and requires several lemmas. The idea is to show the nonexistence of an isometric immersion

φ = ψ ∘ exp p : S ′ ⟶ R 3 {\displaystyle \varphi =\psi \circ \exp _{p}:S'\longrightarrow \mathbb {R} ^{3}}

of a plane S ′ {\displaystyle S'} to the real space R 3 {\displaystyle \mathbb {R} ^{3}} . This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and Spivak. Observations: In order to have a more manageable treatment, but without loss of generality, the curvature may be considered equal to minus one, K = − 1 {\displaystyle K=-1} . There is no loss of generality, since it is being dealt with constant curvatures, and similarities of R 3 {\displaystyle \mathbb {R} ^{3}} multiply K {\displaystyle K} by a constant. The exponential map exp p : T p ( S ) ⟶ S {\displaystyle \exp _{p}:T_{p}(S)\longrightarrow S} is a local diffeomorphism (in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an inner product in the tangent space of S {\displaystyle S} at p {\displaystyle p} : T p ( S ) {\displaystyle T_{p}(S)} . Furthermore, S ′ {\displaystyle S'} denotes the geometric surface T p ( S ) {\displaystyle T_{p}(S)} with this inner product. If ψ : S ⟶ R 3 {\displaystyle \psi :S\longrightarrow \mathbb {R} ^{3}} is an isometric immersion, the same holds for

φ = ψ ∘ exp o : S ′ ⟶ R 3 {\displaystyle \varphi =\psi \circ \exp _{o}:S'\longrightarrow \mathbb {R} ^{3}} . The first lemma is independent from the other ones, and will be used at the end as the counter statement to reject the results from the other lemmas. Lemma 1: The area of S ′ {\displaystyle S'} is infinite. Proof's Sketch: The idea of the proof is to create a global isometry between H {\displaystyle H} and S ′ {\displaystyle S'} . Then, since H {\displaystyle H} has an infinite area, S ′ {\displaystyle S'} will have it too. The fact that the hyperbolic plane H {\displaystyle H} has an infinite area comes by computing the surface integral with the corresponding coefficients of the First fundamental form. To obtain these ones, the hyperbolic plane can be defined as the plane with the following inner product around a point q ∈ R 2 {\displaystyle q\in \mathbb {R} ^{2}} with coordinates ( u , v ) {\displaystyle (u,v)}

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Worked examples

Example 1 — a first encounter with Hilbert's theorem (differential geometry)

Start with the simplest possible case. Write down what Hilbert's theorem (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 Hilbert's theorem (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 Hilbert's theorem (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 Hilbert's theorem (differential geometry)

In research: Hilbert's theorem (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 Hilbert's theorem (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: Hilbert's theorem (differential geometry) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hyperbolic manifolds, Theorems in differential geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Hilbert's theorem (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 Hilbert's theorem (differential geometry) in 20 minutes

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

Frequently asked questions

What is Hilbert's theorem (differential geometry) in simple terms?

Why does Hilbert's theorem (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 Hilbert's theorem (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 Hilbert's theorem (differential geometry).