differential geometry w notes from teacher_Part_16

# differential geometry w notes from teacher_Part_16 - 2.1...

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2.1. COVECTORS AND RIEMANNIAN METRIC 31 Thus, the vector spaces E and E * are isomorphic. The isomorphism is pro- vided by the inner product (the metric). The space E can also be considered as the space of linear functionals on E * . A vector v E deﬁnes a linear functional v : E * R by, for any α E * v ( α ) = α ( v ) . 2.1.4 Riemannian Manifolds Deﬁnition 2.1.6 Let M be a manifold. A Riemannian metric on M is a di erentiable assignment of a positive deﬁnite inner product in each tangent space T p M to the manifold at each point p M. If the inner product is non-degenerate but not positive deﬁnite, then it is a pseudo-Riemannian metric . A Riemannian manifold is a pair ( M , g ) of a manifold with a Rieman- nian metric on it. Let p M be a point in M and ( U , x α ) be a local coordinate system about p . Let i be the coordinate basis in T p M and g α i j = h α i , ∂ α j i be the components of the metric tensor in the coordinate system

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differential geometry w notes from teacher_Part_16 - 2.1...

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