Relate the concept of a tangent vector to a vector field. Prove that the set of all vector fields on a manifold M forms a vector space over C^(M). From the local coordinate approac... Relate the concept of a tangent vector to a vector field. Prove that the set of all vector fields on a manifold M forms a vector space over C^(M). From the local coordinate approach to a differentiable manifold of dimension n, prove that a two-dimensional sphere is an orientable manifold. Explain the notion of alternating covariant and contravariant tensors or order 'r' with reference to the permutation group of naturals leading to the symmetrizing and an alternating map. Define an exterior r-vector and an r-form. Compute the exterior product. Prove that two connections D and D̄ are equal iff they have same geodesics and same torsion tensors. State and prove any two properties of The Lie derivation.
Understand the Problem
The question appears to be from a mathematics exam focused on the topic of differentiable manifolds. It includes several sub-questions related to vector fields, differentiable manifolds, tensor notation, and the properties of connections in differential geometry. The main request is likely seeking help with these concepts or specific proofs.
Answer
Tangent vectors define vector fields; vector fields form a vector space. Spheres are orientable. Tensors relate to permutations. Exterior products involve antisymmetrization. Connections are equal if geodesics, torsion are same. Lie derivatives are linear, respect brackets.
1(a) A tangent vector at each point of a manifold forms a vector field by smoothly assigning these vectors throughout the manifold.
1(b) Vector fields form a vector space over functions on the manifold satisfying vector space axioms.
- Use a consistent choice of orientation via atlas.
3(a) Alternating tensors relate to permutation groups for their antisymmetrization.
3(b) Exterior r-vectors are antisymmetric tensors, and r-forms are differential forms; compute exterior product via distributive properties.
4(a) Connections D and D̄ are equal if they define same geodesics and torsion tensors.
4(b) Lie derivative properties include nilpotency and Leibniz rule.
Answer for screen readers
1(a) A tangent vector at each point of a manifold forms a vector field by smoothly assigning these vectors throughout the manifold.
1(b) Vector fields form a vector space over functions on the manifold satisfying vector space axioms.
- Use a consistent choice of orientation via atlas.
3(a) Alternating tensors relate to permutation groups for their antisymmetrization.
3(b) Exterior r-vectors are antisymmetric tensors, and r-forms are differential forms; compute exterior product via distributive properties.
4(a) Connections D and D̄ are equal if they define same geodesics and torsion tensors.
4(b) Lie derivative properties include nilpotency and Leibniz rule.
More Information
Vector fields on a manifold are critical in differential topology and geometry. They provide both a geometric and algebraic way to understand the manifold's structure.
Tips
Common mistakes involve forgetting w.r.t. coordinate transformations or not verifying the closure properties for vector spaces.
Sources
- Differentiable manifold - Wikipedia - en.wikipedia.org
- [PDF] Differential Geometry - Jay Havaldar - jhavaldar.github.io
- Manifolds, Tangent Spaces, Cotangent - UPenn CIS - cis.upenn.edu
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