An Introduction to Differentiable Manifolds and Riemannian GeometryAcademic Press, 1986/04/21 - 429 ページ An Introduction to Differentiable Manifolds and Riemannian Geometry |
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9 ページ
... suppose q.(p) = x. Then it is clear that interior to any neighborhood V of p there is a neighborhood W whose closure W is in U and for which p(W) = B,(x) with e > 3 > 0. It follows that M is locally connected at p since B,(x) and hence ...
... suppose q.(p) = x. Then it is clear that interior to any neighborhood V of p there is a neighborhood W whose closure W is in U and for which p(W) = B,(x) with e > 3 > 0. It follows that M is locally connected at p since B,(x) and hence ...
22 ページ
... suppose that f is a differentiable curve and maps (a, b) into U, an open subset of R". Let a 3 to < b and suppose that g is a function on U which is differentiable at f(to) e U. Then the composite function g o f is a real-valued ...
... suppose that f is a differentiable curve and maps (a, b) into U, an open subset of R". Let a 3 to < b and suppose that g is a function on U which is differentiable at f(to) e U. Then the composite function g o f is a real-valued ...
23 ページ
... suppose that U is starlike with respect to a. Then given xe U there exists () e R, 0 < 0 < 1, such that wo-oo-j (::), -2) Proof Set f(t) = a + tsz – a), that is x'(t) = a + t|x' — a'). Then the corresponding curve is a line segment with ...
... suppose that U is starlike with respect to a. Then given xe U there exists () e R, 0 < 0 < 1, such that wo-oo-j (::), -2) Proof Set f(t) = a + tsz – a), that is x'(t) = a + t|x' — a'). Then the corresponding curve is a line segment with ...
27 ページ
... suppose U c R" and V c R" are open sets and F: U → V c R" and G: V → RP so that H = Go F is defined on U, which it maps into R*. We may write the coordinate functions of H using those of F and G: h'(x) = g o F(x) = g(f'(x),..., f"(x)) ...
... suppose U c R" and V c R" are open sets and F: U → V c R" and G: V → RP so that H = Go F is defined on U, which it maps into R*. We may write the coordinate functions of H using those of F and G: h'(x) = g o F(x) = g(f'(x),..., f"(x)) ...
33 ページ
... Suppose then y, o, fle R, De 9(a), and f, ge C*(a). Then (yD)(of + £g) = y[D(of + £g)] (by definition of yB) = y[o:(Df) + £(Dg)] (by property (i)) = y.o.(Df) + y£(Dg) (by the distributive law of R) = x(yD)f + £(yD)g (by ...
... Suppose then y, o, fle R, De 9(a), and f, ge C*(a). Then (yD)(of + £g) = y[D(of + £g)] (by definition of yB) = y[o:(Df) + £(Dg)] (by property (i)) = y.o.(Df) + y£(Dg) (by the distributive law of R) = x(yD)f + £(yD)g (by ...
目次
1 | |
20 | |
51 | |
Chapter IV Vector Fields on a Manifold | 106 |
Chapter V Tensors and Tensor Fields on Manifolds | 176 |
Chapter VI Integration on Manifolds | 229 |
Chapter VII Differentiation on Riemannian Manifolds | 297 |
Chapter VIII Curvature | 365 |
References | 417 |
Index | 423 |
多く使われている語句
algebra basis bi-invariant C*-vector field compact completes the proof components connected coordinate frames coordinate neighborhood Corollary corresponding countable covariant tensor covering curve p(t defined definition denote derivative diffeomorphism differentiable manifold dimension domain of integration element equations equivalent Euclidean space example Exercise exists fact finite fixed point formula functions geodesic geometry given Gl(n hence homeomorphism homotopy identity imbedding inner product integral curve isometry isomorphism Lemma Let F Lie group G linear map mapping F matrix notation obtain one-parameter subgroup one-to-one open set open subset oriented orthogonal orthonormal parameter plane properly discontinuously properties prove rank real numbers regular submanifold Remark Riemannian manifold Riemannian metric Section Show structure submanifold subspace suppose surface symmetric tangent space tangent vector tensor field Theorem Let topology uniquely determined vector field vector space zero