mathematics

Geodesic map

Geodesic map 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 Geodesic map rather than just read about it. In short: In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riemannian manifolds (M, g) and (N, h), a function φ : M → N is said to be a geodesic map if φ is a diffeomorphism of M onto N; and the image under φ of any geodesic arc in M is a geodesic arc in N; and the image under the i…

Key takeaways

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

Reference excerpt

In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riemannian manifolds (M, g) and (N, h), a function φ : M → N is said to be a geodesic map if

φ is a diffeomorphism of M onto N; and the image under φ of any geodesic arc in M is a geodesic arc in N; and the image under the inverse function φ−1 of any geodesic arc in N is a geodesic arc in M.

Examples If (M, g) and (N, h) are both the n-dimensional Euclidean space En with its usual flat metric, then any Euclidean isometry is a geodesic map of En onto itself. Similarly, if (M, g) and (N, h) are both the n-dimensional unit sphere Sn with its usual round metric, then any isometry of the sphere is a geodesic map of Sn onto itself. If (M, g) is the unit sphere Sn with its usual round metric and (N, h) is the sphere of radius 2 with its usual round metric, both thought of as subsets of the ambient coordinate space Rn+1, then the "expansion" map φ : Rn+1 → Rn+1 given by φ(x) = 2x induces a geodesic map of M onto N. There is no geodesic map from the Euclidean space En onto the unit sphere Sn, since they are not homeomorphic, let alone diffeomorphic. The gnomonic projection of the hemisphere to the plane is a geodesic map as it takes great circles to lines and its inverse takes lines to great circles. Let (D, g) be the unit disc D ⊂ R2 equipped with the Euclidean metric, and let (D, h) be the same disc equipped with a hyperbolic metric as in the Poincaré disc model of hyperbolic geometry. Then, although the two structures are diffeomorphic via the identity map i : D → D, i is not a geodesic map, since g-geodesics are always straight lines in R2, whereas h-geodesics can be curved. On the other hand, when the hyperbolic metric on D is given by the Klein model, the identity i : D → D is a geodesic map, because hyperbolic geodesics in the Klein model are (Euclidean) straight line segments.

References Ambartzumian, R. V. (1982). Combinatorial integral geometry. Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics. New York: John Wiley & Sons Inc. pp. xvii+221. ISBN 0-471-27977-3. MR 0679133. Kreyszig, Erwin (1991). Differential geometry. New York: Dover Publications Inc. pp. xiv+352. ISBN 0-486-66721-9. MR 1118149.

External links Weisstein, Eric W. "Geodesic mapping". MathWorld.

Worked examples

Example 1 — a first encounter with Geodesic map

Start with the simplest possible case. Write down what Geodesic map 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 Geodesic map 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 Geodesic map 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 Geodesic map

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

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

Frequently asked questions

What is Geodesic map in simple terms?

Why does Geodesic map 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 Geodesic map?

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 Geodesic map.