Learned by Reading
In Wikipedia:
Thus “B” is a positive definite symmetric bilinear form, in other words, an inner product on “V”.
(Parse)
(S1 (S (ADVP (RB Thus)) ('' '')
(NP (SYM B)) ('' '')
(VP (AUX is)
(NP (NP (DT a) (JJ positive) (JJ definite) (JJ symmetric) (NN bilinear) (NN form)) (, ,)
(PP (IN in)
(NP (JJ other) (NNS words)))
(, ,)
(NP (NP (DT an) (JJ inner) (NN product))
(PP (IN on) ('' '')
(NP (NNP V) ('' '')))
(. .)))
In Wikipedia:
Riemannian geometry studies Riemannian manifolds, smooth manifolds with a “Riemannian metric”, a notion of a distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point.
(Parse)
(S1 (S (NP (JJ Riemannian) (NN geometry))
(VP (VBZ studies)
(SBAR (S (NP (NNP Riemannian))
(VP (VBZ manifolds) (PRN (, ,)
(S (NP (NP (JJ smooth) (NNS manifolds))
(PP (IN with)
(NP (DT a) ('' '') (NNP Riemannian))))
(ADJP (JJ metric) ('' '')))
(, ,))
(NP (NP (DT a) (NN notion))
(PP (IN of)
(NP (NP (DT a) (NN distance))
(VP (VBN expressed)
(PP (IN by)
(NP (NP (NNS means))
(PP (IN of)
(NP (NP (DT a) (JJ smooth) (JJ positive) (JJ definite) (JJ symmetric) (NN bilinear) (NN form))
(VP (VBN defined)
(PP (IN on)
(NP (DT the) (JJ tangent) (NN space)))
(PP (IN at)
(NP (DT each) (NN point))))
(. .)))
In Wikipedia:
Such an involution is called a “’Cartan involution ’” on [Formula 1] if [Formula 7] is a positive definite bilinear form.
(Parse)
(S1 (S (NP (PDT Such) (DT an) (NN involution))
(VP (AUX is)
(VP (VBN called)
(S (NP (NP (DT a) ('' '') (NN 'Cartan) (NN involution) ('' ')) ('' '')
(PP (IN on)
(NP (JJ -LSB-) (NN Formula))))
(ADJP (CD 1) (NN -RSB-)))
(SBAR (IN if)
(S (NP (NP (JJ -LSB-) (NN Formula))
(NP (CD 7) (NN -RSB-)))
(VP (AUX is)
(NP (DT a) (JJ positive) (JJ definite) (NN bilinear) (NN form))))
(. .)))