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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2018 • Question 2

Question

Math 6510 • Differentiable Manifolds Summer 2018 • Question 2

Let \(M\) be a smooth manifold containing a point \(p\), and let \(C_p^\infty (M)\) be the germs of smooth real-valued functions on \(M\) at the point \(p\). Prove, using only the definition of a derivation, that \(\delta ([\lambda ])=0\) for any constant \(\lambda \in \mathbb {R}\) and any derivation \(\delta :C_p^\infty (M)\to \mathbb {R}\).

Raw LaTeX

Let \(M\) be a smooth manifold containing a point \(p\), and let
\(C_p^\infty(M)\) be the germs of smooth real-valued functions on \(M\) at the point
\(p\). Prove, using only the definition of a derivation, that
\(\delta([\lambda])=0\) for any constant \(\lambda\in\R\) and any derivation
\(\delta:C_p^\infty(M)\to\R\).

Topic Tags

Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds