1951508
미분기하학
1946966
2010-01-26T20:36:43Z
http://pythagoras0.springnote.com/pages/1951508
2008-10-18T10:09:48Z
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소</h5>
<ul>
<li><a href="/pages/1951508" title="미분기하학" class="wiki">미분기하학</a></li>
</ul>
<p> </p>
<p> </p>
<h5>개요</h5>
<ul>
<li>위상적으로는 국소적으로 유클리드 공간과 같으며, 그 위에 메트릭이 주어진 곡면의 기하학을 공부함.</li>
<li><a href="/pages/3064880" title="비유클리드 기하학" class="wiki">비유클리드기하학</a>을 이해하는 틀을 배우게 된다</li>
<li>미분형식을 이용하여 전개할 수도 있고, 미분형식을 사용하지 않고 크리스토펠 기호를 사용하여 전개할 수도 있다</li>
</ul>
<p> </p>
<p> </p>
<h5>선수 과목 또는 알고 있으면 좋은 것들</h5>
<ul>
<li>
<p><a href="/pages/1942998" class="wiki" title="다변수미적분학">다변수미적분학</a></p>
<ul>
<li>매개화된 곡면</li>
</ul>
</li>
<li><a href="/pages/1953166" class="wiki" title="상미분방정식">상미분방정식</a></li>
<li>기초적인 편미분방정식</li>
<li>
<p><a href="/pages/1932518" class="wiki" title="선형대수학">선형대수학</a></p>
<ul>
<li>내적공간</li>
<li>대칭행렬의 대각화</li>
</ul>
</li>
</ul>
<p> </p>
<p> </p>
<h5>다루는 대상</h5>
<ul>
<li>곡선</li>
<li>곡면</li>
</ul>
<p> </p>
<h5>중요한 개념 및 정리</h5>
<ul>
<li>
<p>메트릭이 주어진 곡면</p>
<ul>
<li>제1기본형식(first fundamental form)</li>
</ul>
</li>
<li><a href="/pages/5082667" class="wiki" title="접속(connection)">접속 (connection)</a></li>
<li>공변미분(covariant derivative)</li>
<li>측지선</li>
<li>
<p>평행이동</p>
<ul>
<li>공변미분이 주어지면, 곡선을 따라 벡터를 평행이동할 수 있다</li>
</ul>
</li>
<li>가우스 곡률</li>
<li><a href="/pages/search?q=%EA%B0%80%EC%9A%B0%EC%8A%A4%EC%9D%98%20%EB%86%80%EB%9D%BC%EC%9A%B4%20%EC%A0%95%EB%A6%AC%28Theorema%20Egregium%29&parent_id=1951508" title="가우스의 놀라운 정리(Theorema Egregium)" class="wiki">가우스의 놀라운 정리(Theorema Egregium)</a></li>
<li><a href="/pages/2628802" class="wiki" title="가우스-보네 정리">가우스-보네 정리</a></li>
</ul>
<p> </p>
<p> </p>
<h5>곡면을 이해하는 두 가지 관점</h5>
<ul>
<li>3차원 공간에 놓인 곡면</li>
<li>3차원 공간을 생각하지 않고, 거리와 각도를 잴 수 있는 방식이 주어진 곡면</li>
<li>학부 미분기하학에서는 곡면에 대한 첫번째 관점에서 두번째 관점으로의 이동을 배우게 됨</li>
</ul>
<p> </p>
<p> </p>
<h5>사용되는 언어</h5>
<ul>
<li>텐서 해석학</li>
<li>미분형식</li>
</ul>
<p> </p>
<p> </p>
<h5>매개화된 곡면</h5>
<ul>
<li><img class="equation" src="http://eq.springnote.com/tex_image?source=X%28u%2Cv%29" alt="X(u,v)" /></li>
<li>벡터장</li>
<li><img class="equation" src="http://eq.springnote.com/tex_image?source=X_1%3A%3DX_u" alt="X_1:=X_u" />, <img class="equation" src="http://eq.springnote.com/tex_image?source=X_2%3A%3DX_v" alt="X_2:=X_v" /></li>
</ul>
<p> </p>
<p> </p>
<h5>제1기본형식(first fundamental form)과 면적소</h5>
<ul>
<li>곡면의 제1기본형식은 곡면 상에서 거리와 각도를 재는 방법에 해당한다</li>
<li>곡면의 접평면마다 내적이 주어진 것으로 이해할 수 있다</li>
<li>메트릭 텐서라고도 한다</li>
<li>
<p>제1기본형식</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=ds%5E2%20%3D%28X_u%20du%2BX_v%20dv%29%5Ccdot%20%28X_u%20du%2BX_v%20dv%29%3D%20Edu%5E2%2B2Fdudv%2BGdv%5E2" alt="ds^2 =(X_u du+X_v dv)\cdot (X_u du+X_v dv)= Edu^2+2Fdudv+Gdv^2" /></p>
</li>
<li>
<p>면적소</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=dA%3D%5Csqrt%7BEG-F%5E2%7D%20%5C%2C%20du%5C%2C%20dv" alt="dA=\sqrt{EG-F^2} \, du\, dv" /></p>
</li>
<li>
<p>3차원의 곡면에 대해서는 다음과 같이 계산할 수 있다</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=g_%7Bij%7D%20%3A%20%3D%20X_i%20%5Ccdot%20X_j" alt="g_{ij} : = X_i \cdot X_j" /></p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=E%3Dg_%7B11%7D" alt="E=g_{11}" />, <img class="equation" src="http://eq.springnote.com/tex_image?source=F%3Dg_%7B12%7D%3Dg_%7B21%7D" alt="F=g_{12}=g_{21}" />, <img class="equation" src="http://eq.springnote.com/tex_image?source=G%3Dg_%7B22%7D" alt="G=g_{22}" /></p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=ds%5E2%20%3D%28X_u%20du%2BX_v%20dv%29%5Ccdot%20%28X_u%20du%2BX_v%20dv%29%3D%20Edu%5E2%2B2Fdudv%2BGdv%5E2" alt="ds^2 =(X_u du+X_v dv)\cdot (X_u du+X_v dv)= Edu^2+2Fdudv+Gdv^2" /></p>
</li>
</ul>
<p> </p>
<p> </p>
<h5>접속(connection)</h5>
<ul>
<li>방향미분의 일반화</li>
<li>벡터장 <img class="equation" src="http://eq.springnote.com/tex_image?source=%7B%5Cmathbf%20v%7D" alt="{\mathbf v}" />와 벡터장 <img class="equation" src="http://eq.springnote.com/tex_image?source=%20%7B%5Cmathbf%20Y%7D" alt=" {\mathbf Y}" />에 대해서 정의되며, 벡터장 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cnabla_%7B%5Cmathbf%20v%7D%20%7B%5Cmathbf%20Y%7D" alt="\nabla_{\mathbf v} {\mathbf Y}" /> 을 얻는다</li>
<li>
<p>다음 성질을 가진다</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cbegin%7Balign%7D%26%5Cnabla_X%28Y_1%20%2B%20Y_2%29%20%3D%20%5Cnabla_XY_1%20%2B%20%5Cnabla_XY_2%5C%5C%0A%26%5Cnabla_%7BX_1%20%2B%20X_2%7DY%20%3D%20%5Cnabla_%7BX_1%7DY%20%2B%20%5Cnabla_%7BX_2%7DY%5C%5C%0A%26%5Cnabla_%7BX%7D%28fY%29%20%3D%20f%5Cnabla_XY%20%2B%20X%28f%29Y%5C%5C%0A%26%5Cnabla_%7BfX%7DY%20%3D%20f%5Cnabla_XY%5Cend%7Balign%7D" alt="\begin{align}&\nabla_X(Y_1 + Y_2) = \nabla_XY_1 + \nabla_XY_2\ &\nabla_{X_1 + X_2}Y = \nabla_{X_1}Y + \nabla_{X_2}Y\ &\nabla_{X}(fY) = f\nabla_XY + X(f)Y\ &\nabla_{fX}Y = f\nabla_XY\end{align}" /></p>
</li>
<li>
<p style="margin: 0px; line-height: 2em;">적당한 1-form <img class="equation" src="http://eq.springnote.com/tex_image?source=A_%7Bij%7D" alt="A_{ij}" style="border-style: none; border-width: 0px; line-height: 2em;" />에 대하여, 다음과 같이 표현할수 있다</p>
<p style="margin: 0px; line-height: 2em;"><img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cnabla%20X_i%20%3D%20%5Csum_%7Bj%3D1%7D%5E%7B2%7D%20A_%7Bij%7D%5Cotimes%20X_j%3D%20A_%7Bi%7D%5E%7Bj%7D%5Cotimes%20X_j" alt="\nabla X_i = \sum_{j=1}^{2} A_{ij}\otimes X_j= A_{i}^{j}\otimes X_j" /></p>
<p style="margin: 0px; line-height: 2em;"><img class="equation" src="http://eq.springnote.com/tex_image?source=A_%7Bij%7D%3D%20A_%7Bi%7D%5E%7Bj%7D" alt="A_{ij}= A_{i}^{j}" /> 로 두었다</p>
</li>
<li>
<p style="margin: 0px; line-height: 2em;">여기서 1-form <img class="equation" src="http://eq.springnote.com/tex_image?source=A_%7Bij%7D" alt="A_{ij}" style="border-style: none; border-width: 0px; line-height: 2em;" />는 벡터장 <img class="equation" src="http://eq.springnote.com/tex_image?source=%7B%5Cmathbf%20v%7D" alt="{\mathbf v}" style="border-style: none; border-width: 0px; line-height: 2em;" />에 대하여 다음을 만족시킴</p>
<p style="margin: 0px; line-height: 2em;"><img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cnabla_%7B%5Cmathbf%20v%7D%20X_i%20%3D%20%28%5Cnabla%20X_i%29%28%7B%5Cmathbf%20v%7D%29%3D%20%5Csum_%7Bj%3D1%7D%5E%7B2%7D%20A_%7Bij%7D%28%7B%5Cmathbf%20v%7D%29%20X_j" alt="\nabla_{\mathbf v} X_i = (\nabla X_i)({\mathbf v})= \sum_{j=1}^{2} A_{ij}({\mathbf v}) X_j" /></p>
</li>
<li>
<p style="margin: 0px; line-height: 2em;">이때의 <img class="equation" src="http://eq.springnote.com/tex_image?source=A%3D%28A_%7Bij%7D%29" alt="A=(A_{ij})" /> 를 접속 1형식(1-form)이라고 부른다</p>
</li>
<li>
<p style="margin: 0px; line-height: 2em;"><img class="equation" src="http://eq.springnote.com/tex_image?source=F%3DdA-A%5Cwedge%20A" alt="F=dA-A\wedge A" /> 는 곡률 2형식(2-form) 이라 부른다</p>
</li>
<li>3차원의 매개화된 곡면의 경우, 접속은 벡터장의 방향미분을 취한뒤, 곡면에 수직한 성분을 빼주는 것으로 얻어진다</li>
<li><a href="/pages/5082667" class="wiki" title="접속(connection)">접속(connection)</a> 항목 참조</li>
</ul>
<p> </p>
<p> </p>
<p> </p>
<h5>크리스토펠 기호</h5>
<ul>
<li>
<p>정의된 접속형식으로부터 다음과 같이 크리스토펠 기호 <img class="equation" src="http://eq.springnote.com/tex_image?source=%7B%5CGamma%5Ek%7D_%7Bij%7D" alt="{\Gamma^k}_{ij}" />, <img class="equation" src="http://eq.springnote.com/tex_image?source=i%2Cj%2Ck%5Cin%5C%7B1%2C2%5C%7D" alt="i,j,k\in\{1,2\}" />를 정의한다</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cnabla_iX_j%20%3D%20%7B%5CGamma%5Ek%7D_%7Bij%7DX_k" alt="\nabla_iX_j = {\Gamma^k}_{ij}X_k" /></p>
</li>
<li>
<p>접속형식 <img class="equation" src="http://eq.springnote.com/tex_image?source=A%3D%28A_%7Bij%7D%29" alt="A=(A_{ij})" style="border-style: none; border-width: 0px; line-height: 2em;" />을 통해서는 다음과 같이 표현할 수 있다</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cnabla_i%20X_j%20%3D%20A_%7Bj%7D%5E%7Bk%7D%28X_i%29%20X_k" alt="\nabla_i X_j = A_{j}^{k}(X_i) X_k" /></p>
<p>즉 <img class="equation" src="http://eq.springnote.com/tex_image?source=%20A_%7Bjk%7D%28X_i%29%3D%7B%5CGamma%5Ek%7D_%7Bij%7D" alt=" A_{jk}(X_i)={\Gamma^k}_{ij}" /></p>
</li>
<li>
<p>3차원 상의 매개화된 곡면의 경우에는 다음과 같이 얻어진다(아래의 *는 곡면에 수직한 성분을 뜻함)</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=X_%7Buu%7D%3D%5CGamma%5E1_%7B11%7DX_u%2B%5CGamma%5E2_%7B11%7DX_v%2B%28*%29" alt="X_{uu}=\Gamma^1_{11}X_u+\Gamma^2_{11}X_v+(*)" /></p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=X_%7Buv%7D%3D%5CGamma%5E1_%7B12%7DX_u%2B%5CGamma%5E2_%7B12%7DX_v%2B%28*%29" alt="X_{uv}=\Gamma^1_{12}X_u+\Gamma^2_{12}X_v+(*)" /></p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=X_%7Bvu%7D%3D%5CGamma%5E1_%7B21%7DX_u%2B%5CGamma%5E2_%7B21%7DX_v%2B%28*%29" alt="X_{vu}=\Gamma^1_{21}X_u+\Gamma^2_{21}X_v+(*)" /></p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=X_%7Bvv%7D%3D%5CGamma%5E1_%7B22%7DX_u%2B%5CGamma%5E2_%7B22%7DX_v%2B%28*%29" alt="X_{vv}=\Gamma^1_{22}X_u+\Gamma^2_{22}X_v+(*)" /></p>
</li>
<li>제1기본형식을 이용한 표현</li>
<li><a href="/pages/4976123" title="크리스토펠 기호" class="wiki">크리스토펠 기호</a> 항목 참조</li>
</ul>
<p> </p>
<p> </p>
<h5>공변미분(covariant derivative), 평행이동, 측지선</h5>
<ul>
<li>
<p>곡선 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Calpha%20%3A%20I%20%5Cto%20M" alt="\alpha : I \to M" /> 를 따라 정의된 벡터장 <img class="equation" src="http://eq.springnote.com/tex_image?source=Y" alt="Y" />에 대하여, 공변미분은 다음과 같이 정의됨</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cfrac%7BDY%7D%7Bdt%7D%3D%5Cnabla_%7B%5Calpha%27%28t%29%7DY" alt="\frac{DY}{dt}=\nabla_{\alpha'(t)}Y" /></p>
</li>
<li>곡선 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Calpha%20%3A%20I%20%5Cto%20M" alt="\alpha : I \to M" style="border-style: none; border-width: 0px; line-height: 2em;" /> 를 따라 정의된 벡터장 <img class="equation" src="http://eq.springnote.com/tex_image?source=Y" alt="Y" style="border-style: none; border-width: 0px; line-height: 2em;" />에 대하여, 공변미분이 0이 되는 경우, 즉 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cnabla_%7B%5Calpha%27%28t%29%7DY%3D0" alt="\nabla_{\alpha'(t)}Y=0" /> 인 경우, 벡터장 <img class="equation" src="http://eq.springnote.com/tex_image?source=Y" alt="Y" style="border-style: none; border-width: 0px; line-height: 2em;" />는 곡선 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Calpha" alt="\alpha" style="border-style: none; border-width: 0px; line-height: 2em;" />를 따라 평행하다고 한다</li>
<li>곡선 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Calpha%20%3A%20I%20%5Cto%20M" alt="\alpha : I \to M" style="border-style: none; border-width: 0px; line-height: 2em;" />가 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cnabla_%7B%5Calpha%27%28t%29%7D%5Calpha%27%28t%29%3D0" alt="\nabla_{\alpha'(t)}\alpha'(t)=0" />를 만족시키는 경우, 곡선 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Calpha" alt="\alpha" style="border-style: none; border-width: 0px; line-height: 2em;" />를 이 곡면의 측지선이라 부른다</li>
<li>
<p>coordinate chart 에서 <img class="equation" src="http://eq.springnote.com/tex_image?source=%5Calpha%28t%29%3D%28%5Calpha_1%28t%29%2C%5Calpha_2%28t%29%29" alt="\alpha(t)=(\alpha_1(t),\alpha_2(t))" style="border-style: none; border-width: 0px; line-height: 2em;" /> 로 표현되는 경우, 크리스토펠 기호를 쓰면 측지선은 다음 미분방정식을 만족시킨다</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=%5Cfrac%7Bd%5E2%5Calpha_k%20%7D%7Bdt%5E2%7D%20%2B%20%5CGamma%5E%7Bk%7D_%7B%7Ei%20j%20%7D%5Cfrac%7Bd%5Calpha_i%20%7D%7Bdt%7D%5Cfrac%7Bd%5Calpha_j%20%7D%7Bdt%7D%20%3D%200" alt="\frac{d^2\alpha_k }{dt^2} + \Gamma^{k}_{~i j }\frac{d\alpha_i }{dt}\frac{d\alpha_j }{dt} = 0" /></p>
</li>
<li><a href="/pages/5063813" title="측지선" class="wiki">측지선</a></li>
</ul>
<p> </p>
<p> </p>
<p> </p>
<h5>가우스곡률과 가우스의 정리</h5>
<ul>
<li>곡면에 정의된 가우스곡률은 곡선의 곡률과 마찬가지로, 곡면에 수직인 정규벡터(normal vector)의 변화와 관련된 개념</li>
<li>가우스의 정리는 이러한 곡률을 곡면의 제1기본형식을 통해서 표현할 수 있음을 말해준다</li>
<li>
<p>가우스곡률은 <img class="equation" src="http://eq.springnote.com/tex_image?source=F%3D0" alt="F=0" />인 제1기본형식에 대하여, 다음과 같이 주어진다</p>
<p><img class="equation" src="http://eq.springnote.com/tex_image?source=K%20%3D%20-%5Cfrac%7B1%7D%7B2%5Csqrt%7BEG%7D%7D%5Cleft%28%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20u%7D%5Cfrac%7BG_u%7D%7B%5Csqrt%7BEG%7D%7D%20%2B%20%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20v%7D%5Cfrac%7BE_v%7D%7B%5Csqrt%7BEG%7D%7D%5Cright%29" alt="K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right)" /></p>
</li>
<li>곡률의 개념이 곡면이 3차원에 놓인 상태와는 무관한 곡면에서 측량할 수 있다는 사실을 말해줌</li>
<li><a href="/pages/3782583" class="wiki" title="가우스의 놀라운 정리(Theorema Egregium)">가우스의 놀라운 정리(Theorema Egregium)</a> 항목 참조</li>
<li><a href="/pages/4976643" title="가우스곡률" class="wiki">가우스곡률</a></li>
</ul>
<p> </p>
<p> </p>
<h5>곡면의 예</h5>
<ul>
<li>
<p>유클리드평면, <a href="/pages/4975767" title="구면(sphere)" class="wiki">구면(sphere)</a>, <a href="/pages/4771317" title="푸앵카레 상반평면 모델" class="wiki">푸앵카레 상반평면</a> 세가지 상수 곡률 곡면</p>
<ul>
<li>이들 곡면은 각각 유클리드기하학, 구면기하학, 쌍곡기하학의 모델</li>
</ul>
</li>
<li><a href="http://www.wolframalpha.com/input/?i=pseudosphere">http://www.wolframalpha.com/input/?i=pseudosphere</a></li>
<li><a href="http://www.wolframalpha.com/input/?i=sphere">http://www.wolframalpha.com/input/?i=sphere</a></li>
</ul>
<p> </p>
<p> </p>
<h5>역사</h5>
<ul>
<li><a href="/pages/3304643" title="수학사연표" class="wiki">수학사연표</a></li>
<li>1829 - 볼리아이, 가우스, 로바체프스키가 <a href="/pages/3304643#" title="쌍곡기하학" class="wiki">쌍곡기하학</a>을 발견</li>
<li>1854 - 리만이 리만기하학을 소개</li>
</ul>
<p> </p>
<p> </p>
<h5>메모</h5>
<ul>
<li><a href="http://math.uh.edu/%7Eminru/Riemann09/five1.pdf">http://math.uh.edu/~minru/Riemann09/five1.pdf</a></li>
</ul>
<p> </p>
<p> </p>
<h5>다른 과목과의 관련성</h5>
<ul>
<li>
<p><a href="/pages/1954084" class="wiki" title="대수적 위상수학">대수적 위상수학</a></p>
<ul>
<li>
<p>오일러의 정리 V-E+F = 2-2g</p>
<ul>
<li>g = 곡면의 종수(genus), 쉽게 말하면 구멍의 개수. 구면은 g=0, 도넛모양은 g=1</li>
<li>가우스-보네 정리를 이해하는데 필수적인 개념</li>
</ul>
</li>
</ul>
</li>
<li>
<p><a href="/pages/1949486" class="wiki" title="복소함수론">복소함수론</a></p>
<ul>
<li>단위원 또는 <a href="/pages/4771317" class="wiki" title="푸앵카레 상반평면 모델">푸앵카레 상반평면 모델</a>의 group of conformal automorphisms = group of isometries</li>
<li><a href="/pages/search?q=Uniformization%20theorem&parent_id=1951508" title="Uniformization theorem" class="wiki">Uniformization theorem</a></li>
</ul>
</li>
<li>
<p><a href="/pages/1940206" class="wiki" title="추상대수학">추상대수학</a></p>
<ul>
<li>군론 - discrete subgroups of isometry group</li>
<li>클라인의 에를랑겐 프로그램(Erlangen Program)</li>
</ul>
</li>
<li>
<p><a href="/pages/1944496" title="미분형식 (differential forms)과 multilinear algebra" class="wiki">미분형식 (differential forms)과 multilinear algebra</a></p>
<ul>
<li>다변수미적분학과 미분기하학을 고차원의 다양체로 확장하기 위해 필요한 언어</li>
</ul>
</li>
</ul>
<p> </p>
<p> </p>
<h5>관련된 대학원 과목 또는 더 공부하면 좋은 것들</h5>
<ul>
<li>
<p>미분다양체론(differentiable manifolds)</p>
<ul>
<li>다양체란 1차원 공간인 곡선, 2차원 공간인 곡면을 일반화한 n차원의 공간</li>
<li>미분다양체는 미적분학을 할 수 있는 다양체를 뜻함</li>
</ul>
</li>
<li>
<p>리만기하학(Riemannian geometry)</p>
<ul>
<li>곡면에 메트릭을 주는 것을 일반화하여 메트릭이 주어진 미분다양체를 공부함</li>
</ul>
</li>
<li>리군과 Symmetric spaces의 분류</li>
</ul>
<p> </p>
<p> </p>
<h5>유명한 정리 혹은 재미있는 문제</h5>
<ul>
<li><a href="http://bomber0.byus.net/index.php/2008/09/26/785" title="http://bomber0.byus.net/index.php/2008/09/26/785" class="external">구면삼각형의 넓이에 대한 Girard-Harriot의 정리</a></li>
</ul>
<p> </p>
<p> </p>
<h5>관련된 항목들</h5>
<ul>
<li><a href="/pages/3065152" title="구면기하학" class="wiki">구면기하학</a></li>
<li><a href="/pages/3065168" title="쌍곡기하학" class="wiki">쌍곡기하학</a></li>
</ul>
<p> </p>
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5>
<ul>
<li>http://www.google.com/dictionary?langpair=en|ko&q=</li>
<li>
<p style="margin: 0px; line-height: 2em;"><a href="http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr=" class="external" title="http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr=" style="line-height: 2em; color: rgb(86, 137, 66) ! important; text-decoration: underline; cursor: pointer ! important;">대한수학회 수학 학술 용어집</a></p>
<ul>
<li style="line-height: 2em;"><a href="http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=fundamental+form">http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=fundamental+form</a></li>
<li style="line-height: 2em;"><a href="http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=connection">http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=connection</a></li>
</ul>
</li>
<li style="line-height: 2em;"><a href="http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4" title="http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid={D6048897-56F9-43D7-8BB6-50B362D1243A}&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4" class="external" style="line-height: 2em; color: rgb(86, 137, 66) ! important; text-decoration: underline; cursor: pointer ! important;">대한수학회 수학용어한글화 게시판</a></li>
</ul>
<p> </p>
<p> </p>
<h5>사전 형태의 자료</h5>
<ul>
<li><a href="http://ko.wikipedia.org/wiki/%EB%AF%B8%EB%B6%84%EA%B8%B0%ED%95%98%ED%95%99">http://ko.wikipedia.org/wiki/미분기하학</a></li>
<li><a href="http://en.wikipedia.org/wiki/differential_geometry">http://en.wikipedia.org/wiki/differential_geometry</a></li>
<li><a href="http://en.wikipedia.org/wiki/First_fundamental_form">http://en.wikipedia.org/wiki/First_fundamental_form</a></li>
<li><a href="http://en.wikipedia.org/wiki/Second_fundamental_form">http://en.wikipedia.org/wiki/Second_fundamental_form</a></li>
<li><a href="http://en.wikipedia.org/wiki/Christoffel_symbols">http://en.wikipedia.org/wiki/Christoffel_symbols</a></li>
<li><a href="http://en.wikipedia.org/wiki/Covariant_derivative">http://en.wikipedia.org/wiki/Covariant_derivative</a></li>
<li><a href="http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html">http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html</a></li>
<li>http://www.wolframalpha.com/input/?i=</li>
<li><a href="http://dlmf.nist.gov/" class="external" title="http://dlmf.nist.gov">NIST Digital Library of Mathematical Functions</a></li>
</ul>
<p> </p>
<p> </p>
<h5>관련논문</h5>
<ul>
<li>http://www.jstor.org/action/doBasicSearch?Query=</li>
</ul>
<ul>
<li class="title">
<p><a href="http://www.jstor.org/stable/2974765" title="http://www.jstor.org/stable/2974765" class="external">Geometry and the Foucault Pendulum</a></p>
<ul>
<li class="author">John Oprea, <cite>The American Mathematical Monthly</cite>, Vol. 102, No. 6 (Jun. - Jul., 1995), pp. 515-522</li>
</ul>
</li>
<li class="title">
<p><a href="http://www.jstor.org/stable/2319962" title="http://www.jstor.org/stable/2319962" class="external">Kleinian Transformation Geometry</a></p>
<ul>
<li class="author">Richard S. Millman, <cite>The American Mathematical Monthly</cite>, Vol. 84, No. 5 (May, 1977), pp. 338-349</li>
</ul>
</li>
<li class="title">
<p><a href="http://www.jstor.org/stable/2319607" title="http://www.jstor.org/stable/2319607" class="external">The Geometry of Connections</a></p>
<ul>
<li class="author">R. S. Millman and Ann K. Stehney, <cite>The American Mathematical Monthly</cite>, Vol. 80, No. 5 (May, 1973), pp. 475-500</li>
</ul>
</li>
<li class="title">
<p><a href="http://www.jstor.org/stable/2321093" title="http://www.jstor.org/stable/2321093" class="external">From Triangles to Manifolds</a></p>
<ul>
<li class="author">Shing-Shen Chern, <cite>The American Mathematical Monthly</cite>, Vol. 86, No. 5 (May, 1979), pp. 339-349</li>
</ul>
</li>
<li class="title">
<p><a href="http://www.jstor.org/stable/2974912" title="http://www.jstor.org/stable/2974912" class="external">How Hyperbolic Geometry Became Respectable</a></p>
<ul>
<li class="author">Abe Shenitzer, <cite>The American Mathematical Monthly</cite>, Vol. 101, No. 5 (May, 1994), pp. 464-470</li>
</ul>
</li>
</ul>
<p> </p>
<p> </p>
<h5>표준적인 교과서</h5>
<ul>
<li>do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces.</li>
</ul>
<p> </p>
<p> </p>
<h5>추천도서 및 보조교재</h5>
<ul>
<li>
<p class="parseasinTitle"><span id="btAsinTitle" style=""><a href="http://www.amazon.com/Gateway-Modern-Geometry-Poincare-Half-Plane/dp/0763753815/ref=sr_1_1?ie=UTF8&s=books&qid=1259658855&sr=1-1" title="http://www.amazon.com/Gateway-Modern-Geometry-Poincare-Half-Plane/dp/0763753815/ref=sr_1_1?ie=UTF8&s=books&qid=1259658855&sr=1-1" class="external">Poincare Half-Plane (A Gateway to Modern Geometry)</a></span></p>
<ul>
<li>
<p>S. Stahl, Jones & Bartlett Publishers; 2 edition (November 25, 2007)</p>
</li>
</ul>
</li>
<li>
<p><a href="http://www.amazon.com/Geometry-Surfaces-John-Stillwell/dp/0387977430" title="http://www.amazon.com/Geometry-Surfaces-John-Stillwell/dp/0387977430" class="external">Geometry of Surfaces</a></p>
<ul>
<li>John Stillwell</li>
</ul>
</li>
<li>
<p><a href="http://www.amazon.com/Shape-Space-Pure-Applied-Mathematics/dp/0824707095">The Shape of Space</a></p>
<ul>
<li>Jeffrey R. Weeks</li>
<li>일반 독자를 위한 책</li>
</ul>
</li>
</ul>
<p> </p>
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