Tensor bundle

Key takeaways

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

Reference excerpt

In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors.

Definition

A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle. Explicitly, for fixed non-negative integers p {\displaystyle p} and q {\displaystyle q} , a tensor bundle is the fiber bundle

V ⊗ ⋯ ⊗ V ⊗ V ∗ ⊗ ⋯ ⊗ V ∗ = V ⊗ p ⊗ ( V ∗ ) ⊗ q {\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}=V^{\otimes p}\otimes (V^{*})^{\otimes q}}

where V is the tangent bundle of M and V∗ is the cotangent bundle of M, and its fibers are

V x ⊗ ⋯ ⊗ V x ⊗ V x ∗ ⊗ ⋯ ⊗ V x ∗ = V x ⊗ p ⊗ ( V x ∗ ) ⊗ q {\displaystyle V_{x}\otimes \cdots \otimes V_{x}\otimes V_{x}^{*}\otimes \cdots \otimes V_{x}^{*}=V_{x}^{\otimes p}\otimes (V_{x}^{*})^{\otimes q}}

where V x {\displaystyle V_{x}} is the tangent space of x and V x ∗ {\displaystyle V_{x}^{*}} is the cotangent space of x, for every x in M. The elements of the fibers are tensors of type ( p , q ) ∈ N × N {\displaystyle (p,q)\in \mathbb {N} \times \mathbb {N} } . The direct sum of vector bundles allows to consider all these posibilities of tensor bundle for fixed ( p , q ) {\displaystyle (p,q)} as a single mathematical object ⨁ ( p , q ) ∈ N × N V ⊗ p ⊗ ( V ∗ ) ⊗ q . {\displaystyle \bigoplus _{(p,q)\in \mathbb {N} \times \mathbb {N} }V^{\otimes p}\otimes (V^{*})^{\otimes q}.}

References

Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771. Saunders, David J. (1989). The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series. Vol. 142. Cambridge New York: Cambridge University Press. ISBN 978-0-521-36948-0. OCLC 839304386. Steenrod, Norman (5 April 1999). The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-00548-5. OCLC 40734875.

See also Fiber bundle – Continuous surjection satisfying a local triviality condition Spinor bundle – Geometric structure Tensor field – Assignment of a tensor continuously varying across a region of space

Worked examples

Example 1 — a first encounter with Tensor bundle

Start with the simplest possible case. Write down what Tensor bundle 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 Tensor bundle 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 Tensor bundle 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 Tensor bundle

In research: Tensor bundle 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 Tensor bundle 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: Tensor bundle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential geometry stubs, Vector bundles, so understanding it makes those chapters shorter.
In everyday life: Look for Tensor bundle 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.

How to study Tensor bundle in 20 minutes

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

Frequently asked questions

What is Tensor bundle in simple terms?

In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors.

Why does Tensor bundle 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 Tensor bundle?

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 Tensor bundle.