definition of Riemannian metric over \(C^\infty\) manifold with boundary
Topics
About: Riemannian manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of Riemannian metric over \(C^\infty\) manifold with boundary.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( (T^0_2 (TM), M, \pi)\): \(= \text{ the } C^\infty (0, 2) \text{ -tensors bundle over } M\)
\(*g\): \(: M \to T^0_2 (TM)\), \(\in \{\text{ the } C^\infty (p, q) \text{ -tensors fields over } M\}\)
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Conditions:
\(\forall v, v' \in T_mM (g (v, v') = g (v', v))\)
\(\land\)
\(\forall v \in T_mM (0 \le g (v, v) \land (v = 0 \iff g (v, v) = 0))\)
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2: Note
In other words, any Riemannian metric is any \(C^\infty\) \((0, 2)\)-tensors field over \(M\) that is symmetric and positive definite.
