mathematics

Fundamental theorem of Riemannian geometry

Fundamental theorem of Riemannian 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 Fundamental theorem of Riemannian geometry rather than just read about it. In short: The fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given metric. Because it is canonically defined by such properties, this connection is often automatically used when given a metric.

Key takeaways

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

Reference excerpt

The fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given metric. Because it is canonically defined by such properties, this connection is often automatically used when given a metric.

Statement The theorem can be stated as follows:

Fundamental theorem of Riemannian Geometry. Let (M, g) be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection ∇ that satisfies the following conditions: for any vector fields X, Y, and Z we have X ( g ( Y , Z ) ) = g ( ∇ X Y , Z ) + g ( Y , ∇ X Z ) , {\displaystyle X{\big (}g(Y,Z){\big )}=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z),} where X(g(Y, Z)) denotes the derivative of the function g(Y, Z) along vector field X. for any vector fields X, Y, ∇ X Y − ∇ Y X = [ X , Y ] , {\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y],} where [X, Y] denotes the Lie bracket of X and Y. The first condition is called metric-compatibility of ∇. It may be equivalently expressed by saying that, given any curve in M, the inner product of any two ∇–parallel vector fields along the curve is constant. It may also be equivalently phrased as saying that the metric tensor is preserved by parallel transport, which is to say that the metric is parallel when considering the natural extension of ∇ to act on (0,2)-tensor fields: ∇g = 0. It is further equivalent to require that the connection is induced by a principal bundle connection on the orthonormal frame bundle. The second condition is sometimes called symmetry of ∇. It expresses the condition that the torsion of ∇ is zero, and as such is also called torsion-freeness. There are alternative characterizations. An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor. The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called the Levi-Civita connection or (pseudo-)Riemannian connection. However, the existence result is extremely direct, as the connection in question may be explicitly defined by either the second Christoffel identity or Koszul formula as obtained in the proofs below. This explicit definition expresses the Levi-Civita connection in terms of the metric and its first derivatives. As such, if the metric is k-times continuously differentiable, then the Levi-Civita connection is (k − 1)-times continuously differentiable. The Levi-Civita connection can also be characterized in other ways, for instance via the Palatini variation of the Einstein–Hilbert action.

Proof The proof of the theorem can be presented in various ways. Here the proof is first given in the language of coordinates and Christoffel symbols, and then in the coordinate-free language of covariant derivatives. Regardless of the presentation, the idea is to use the metric-compatibility and torsion-freeness conditions to obtain a direct formula for any connection that is both metric-compatible and torsion-free. This establishes the uniqueness claim in the fundamental theorem. To establish the existence claim, it must be directly checked that the formula obtained does define a connection as desired.

Local coordinates Here the Einstein summation convention will be used, which is to say that an index repeated as both subscript and superscript is being summed over all values. Let m denote the dimension of M. Recall that, relative to a local chart, a connection is given by m3 smooth functions

{ Γ l

i j } , {\displaystyle \left\{\Gamma ^{l}{}_{ij}\right\},}

with

( ∇ X Y ) i = X j ∂ j Y i + X j Y k Γ i

j k {\displaystyle (\nabla _{X}Y)^{i}=X^{j}\partial _{j}Y^{i}+X^{j}Y^{k}\Gamma ^{i}{}_{jk}}

for any vector fields X and Y. Torsion-freeness of the connection refers to the condition that ∇XY − ∇Y X = [X, Y] for arbitrary X and Y. Written in terms of local coordinates, this is equivalent to

0 = X j Y k ( Γ i

j k − Γ i

k j ) , {\displaystyle 0=X^{j}Y^{k}(\Gamma ^{i}{}_{jk}-\Gamma ^{i}{}_{kj}),}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Fundamental theorem of Riemannian geometry

Start with the simplest possible case. Write down what Fundamental theorem of Riemannian 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 Fundamental theorem of Riemannian 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 Fundamental theorem of Riemannian 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 Fundamental theorem of Riemannian geometry

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

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

Frequently asked questions

What is Fundamental theorem of Riemannian geometry in simple terms?

Why does Fundamental theorem of Riemannian 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 Fundamental theorem of Riemannian 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 Fundamental theorem of Riemannian geometry.