mathematics

Myers's theorem

Myers's theorem 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 Myers's theorem rather than just read about it. In short: Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941.

Key takeaways

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

Reference excerpt

Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:

In the special case of surfaces, this result was proved by Ossian Bonnet in 1855. For a surface, the Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the sectional curvature. Myers' key contribution was therefore to show that a Ricci lower bound is all that is needed to reach the same conclusion.

Corollaries The conclusion of the theorem says, in particular, that the diameter of ( M , g ) {\displaystyle (M,g)} is finite. Therefore M {\displaystyle M} must be compact, as a closed (and hence compact) ball of finite radius in any tangent space is carried onto all of M {\displaystyle M} by the exponential map. As a very particular case, this shows that any complete and noncompact smooth Einstein manifold must have nonpositive Einstein constant. Since M {\displaystyle M} is connected, there exists the smooth universal covering map π : N → M . {\displaystyle \pi :N\to M.} One may consider the pull-back metric π ∗ g {\displaystyle \pi ^{*}g} on N . {\displaystyle N.} Since π {\displaystyle \pi } is a local isometry, Myers' theorem applies to the Riemannian manifold ( N , π ∗ g ) {\displaystyle (N,\pi ^{*}g)} and hence N {\displaystyle N} is compact and the covering map is finite. This implies that the fundamental group of M {\displaystyle M} is finite.

Cheng's diameter rigidity theorem The conclusion of Myers' theorem says that for any p , q ∈ M , {\displaystyle p,q\in M,} one has d g ( p , q ) ≤ π k {\displaystyle d_{g}(p,q)\leq {\frac {\pi }{\sqrt {k}}}} . In 1975, Shiu-Yuen Cheng proved:

Let ( M , g ) {\displaystyle (M,g)} be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ricg ≥ (n-1)k, and if there exists p and q in M with d g ( p , q ) = π k {\displaystyle d_{g}(p,q)={\frac {\pi }{\sqrt {k}}}} , then ( M , g ) {\displaystyle (M,g)} is simply-connected and has constant sectional curvature k.

See also Gromov's compactness theorem (geometry) – On when a set of compact Riemannian manifolds of a given dimension is relatively compact

References

Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348. Cheng, Shiu Yuen (1975), "Eigenvalue comparison theorems and its geometric applications", Mathematische Zeitschrift, 143 (3): 289–297, doi:10.1007/BF01214381, ISSN 0025-5874, MR 0378001 do Carmo, M. P. (1992), Riemannian Geometry, Boston, Mass.: Birkhäuser, ISBN 0-8176-3490-8 Myers, S. B. (1941), "Riemannian manifolds with positive mean curvature", Duke Mathematical Journal, 8 (2): 401–404, doi:10.1215/S0012-7094-41-00832-3

Worked examples

Example 1 — a first encounter with Myers's theorem

Start with the simplest possible case. Write down what Myers's theorem 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 Myers's theorem 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 Myers's theorem 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 Myers's theorem

In research: Myers's theorem 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 Myers's theorem 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: Myers's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric inequalities, Theorems in Riemannian geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Myers's theorem 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Myers's theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Myers's theorem in 20 minutes

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

Frequently asked questions

What is Myers's theorem in simple terms?

Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941.

Why does Myers's theorem 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 Myers's theorem?

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 Myers's theorem.