differential geometry w notes from teacher_Part_15

# differential geometry w notes from teacher_Part_15 - 2.1...

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Unformatted text preview: 2.1. COVECTORS AND RIEMANNIAN METRIC 29 4. If, in addition, ∀ v ∈ E , h v , v i ≥ and h v , v i = if and only if v = , then the inner product is called positive definite . • For a positive-definite inner product the norm of a vector v is defined by || v || = p h v , v i • Let { e j } be a basis in E . • Then the matrix g ij defined by g ij = h e i , e j i is a metric tensor , more precisely it gives the components of the metric tensor in that basis. • The matrix g ij is symmetric and nondegenerate, that is, g ij = g ji , det g ij , . For a positive definite inner product, this matrix is positive-definite, that is, it has only positive real eigenvalues. One says, that the metric has the signature ( + ··· + ). In special relativity one considers metrics wich are not positive definite but have the signature (- + ··· + ). • Two vectors v , w ∈ E are orthogonal if h v , w i = ....
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differential geometry w notes from teacher_Part_15 - 2.1...

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