Sectional curvature - Revision history

Vipul: /* Facts */

2008-05-22T12:38:45Z

Facts

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	''This is analogous to the fact that [[Ricci curvature]] determines the [[Ricci curvature tensor]]''		''This is analogous to the fact that [[Ricci curvature]] determines the [[Ricci curvature tensor]]''

	~~Given the sectional curvature for all tangent planes, we can back-calculate the Riemann curvature tensor from it by the usual polarization trick.~~ The ~~idea is to view~~ the Riemann curvature tensor a s a symmetric bilinear form on <math>\Lambda^2(TM)</math>, and the sectional curvature as the values that this form takes on pairs <math>(\Pi,\Pi)</math> of ''unit vectors'' in <math>\Lambda^2(\Gamma(TM))</math>. To clarify:		The following facts are true about the Riemann curvature tensor:

	* The [[Riemann curvature tensor of Levi-Civita connection\|Riemann curvature tensor]] can be viewed as a <math>(0,4)</math>-tensor:		* The [[Riemann curvature tensor of Levi-Civita connection\|Riemann curvature tensor]] can be viewed as a <math>(0,4)</math>-tensor:
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	* The above map is antisymmetric in <math>X, Y</math> ({{proofat\|[[Antisymmetry of Riemann curvature tensor]]}} and antisymmetric in <math>Z</math> and <math>W</math>. Further it is symmetric in the pairs <math>(X,Y)</math> and <math>(W,Z)</math>. Hence, it is a symmetric bilinear form on <math>\Lambda^2(\Gamma(TM))</math>.		* The above map is antisymmetric in <math>X, Y</math> ({{proofat\|[[Antisymmetry of Riemann curvature tensor]]}} and antisymmetric in <math>Z</math> and <math>W</math>. Further it is symmetric in the pairs <math>(X,Y)</math> and <math>(W,Z)</math>. Hence, it is a symmetric bilinear form on <math>\Lambda^2(\Gamma(TM))</math>.

	* A unit vector in <math>\Lambda^2(\Gamma(TM))</math> defines on each tangent ~~space~~, ~~an element of~~ <math>\Lambda^2(~~T_pM~~)</math> that has norm 1. The norm of an element in the second exterior power is the square of the area of the parallelogram it spans, so unit norm vectors are things of the form <math>X \wedge Y</math> where <math>X,Y</math> span a parallelogram of area <math>1</math>. Thus, the denominator vanishes for these pairs, and the sectional curvature ~~is simply~~ <math>R(X,~~Y,Y,X~~)</math> ~~which is the inner product~~ of <math>X \~~wedge Y~~</math> ~~with itself~~.		Thus, given the sectional curvature for all tangent planes, we can back-calculate the Riemann curvature tensor from it by the usual polarization trick. The idea is to view the Riemann curvature tensor as a symmetric bilinear form on <math>\Lambda^2(TM)</math>, and the sectional curvature as the values that this form takes on pairs <math>(\Pi,\Pi)</math> of (special kinds of) ''unit vectors'' in <math>\Lambda^2(\Gamma(TM))</math>.

Vipul: 9 revisions

2008-05-18T20:08:56Z

9 revisions

← Older revision	Revision as of 20:08, 18 May 2008
(No difference)

Vipul at 03:59, 31 August 2007

2007-08-31T03:59:51Z

← Older revision		Revision as of 03:59, 31 August 2007
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	* [[Constant-curvature metric]]		* [[Constant-curvature metric]]
	* [[Flat metric]]		* [[Flat metric]]

			==Facts==

			===Sectional curvature determines Riemann curvature tensor===

			''This is analogous to the fact that [[Ricci curvature]] determines the [[Ricci curvature tensor]]''

			Given the sectional curvature for all tangent planes, we can back-calculate the Riemann curvature tensor from it by the usual polarization trick. The idea is to view the Riemann curvature tensor a s a symmetric bilinear form on <math>\Lambda^2(TM)</math>, and the sectional curvature as the values that this form takes on pairs <math>(\Pi,\Pi)</math> of ''unit vectors'' in <math>\Lambda^2(\Gamma(TM))</math>. To clarify:

			* The [[Riemann curvature tensor of Levi-Civita connection\|Riemann curvature tensor]] can be viewed as a <math>(0,4)</math>-tensor:

			<math>R(X,Y,Z,W) = g(R(X,Y)Z,W)</math>

			* The above map is antisymmetric in <math>X, Y</math> ({{proofat\|[[Antisymmetry of Riemann curvature tensor]]}} and antisymmetric in <math>Z</math> and <math>W</math>. Further it is symmetric in the pairs <math>(X,Y)</math> and <math>(W,Z)</math>. Hence, it is a symmetric bilinear form on <math>\Lambda^2(\Gamma(TM))</math>.

			* A unit vector in <math>\Lambda^2(\Gamma(TM))</math> defines on each tangent space, an element of <math>\Lambda^2(T_pM)</math> that has norm 1. The norm of an element in the second exterior power is the square of the area of the parallelogram it spans, so unit norm vectors are things of the form <math>X \wedge Y</math> where <math>X,Y</math> span a parallelogram of area <math>1</math>. Thus, the denominator vanishes for these pairs, and the sectional curvature is simply <math>R(X,Y,Y,X)</math> which is the inner product of <math>X \wedge Y</math> with itself.

Vipul: /* Related metric properties */

2007-08-31T02:59:43Z

Related metric properties

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	* [[Riemannian manifold with positively lower-bounded curvature]]		* [[Riemannian manifold with positively lower-bounded curvature]]
	* [[Positively curved Riemannian manifold]]		* [[Positively curved Riemannian manifold]]
			* [[Quasi-positively curved Riemannian manifold]]
			* [[Nonnegatively curved Riemannian manifold]]
	* [[Negatively curved Riemannian manifold]]		* [[Negatively curved Riemannian manifold]]
	* [[Constant-curvature metric]]		* [[Constant-curvature metric]]
			* [[Flat metric]]

Vipul: /* Definition */

2007-08-31T02:59:02Z

Definition

← Older revision		Revision as of 02:59, 31 August 2007
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	Note that the denominator cannot vanish because <math>X</math> and <math>Y</math> are independent vectors.		Note that the denominator cannot vanish because <math>X</math> and <math>Y</math> are independent vectors.

			===For a pseudo-Riemannian manifold===

			{{further\|[[Sectional curvature for a pseudo-Riemannian manifold]]}}

			We can also define the sectional curvature of a pseudo-Riemannian manifold. The same definition works.

	==Related notions==		==Related notions==

Vipul: /* Definition part */

2007-08-31T02:51:03Z

Definition part

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	Let <math>\Pi</math> be a tangent plane to <math>M</math> at a point <math>p \in M</math>. Then, the sectional curvature of <math>\Pi</math> at <math>p</math> is defined as follows: take two linearly independent vectors <math>X</math> and <math>Y</math> in <math>\Pi</math>, and calculate:		Let <math>\Pi</math> be a tangent plane to <math>M</math> at a point <math>p \in M</math>. Then, the sectional curvature of <math>\Pi</math> at <math>p</math> is defined as follows: take two linearly independent vectors <math>X</math> and <math>Y</math> in <math>\Pi</math>, and calculate:

	<math><R(X,Y)Y,X></math>		<math>g(R(X,Y)Y,X)</math>

	viz the inner product of <math>R(X,Y)Y</math> and <math>X</math> with respect to <math>g</math>.		viz the inner product of <math>R(X,Y)Y</math> and <math>X</math> with respect to <math>g</math>.

	Divide this by the square of the area of the parallelogram formed by <math>X</math> and <math>Y</math>. This ratio defines the sectional curvature of <math>\Pi</math>, denoted as <math>K(\Pi)</math>.		Divide this by the square of the area of the parallelogram formed by <math>X</math> and <math>Y</math>. This ratio defines the sectional curvature of <math>\Pi</math>, denoted as <math>K(\Pi)</math>.

			Here is the more explicit formula:

			<math>\frac{g(R(X,Y)Y,X)}{g(X,X)g(Y,Y) - g(X,Y)^2}</math>

			Note that the denominator cannot vanish because <math>X</math> and <math>Y</math> are independent vectors.

	==Related notions==		==Related notions==

Vipul: /* Related metric properties */

2007-07-07T11:32:08Z

Related metric properties

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	===Related metric properties===		===Related metric properties===

	* [[~~Metric of positive~~ lower-bounded ~~sectional~~ curvature]]		* [[Riemannian manifold with positively lower-bounded curvature]]
	* [[~~Metric of positive sectional curvature~~]]		* [[Positively curved Riemannian manifold]]
	* [[~~Metric of negative sectional curvature~~]]		* [[Negatively curved Riemannian manifold]]
	* [[Constant-curvature metric]]		* [[Constant-curvature metric]]

Vipul at 18:03, 14 June 2007

2007-06-14T18:03:52Z

← Older revision		Revision as of 18:03, 14 June 2007
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	{{curvature notion}}		{{curvature notion}}

	{{pseudo-Riemannian\|[[Sectional curvature of a pseudo-Riemannian manifold]]}}		{{pseudo-Riemannian\|[[Sectional curvature for a pseudo-Riemannian manifold]]}}

	==Definition==		==Definition==

Vipul at 17:59, 14 June 2007

2007-06-14T17:59:29Z

← Older revision		Revision as of 17:59, 14 June 2007
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	{{curvature notion}}		{{curvature notion}}

			{{pseudo-Riemannian\|[[Sectional curvature of a pseudo-Riemannian manifold]]}}

	==Definition==		==Definition==

Vipul: /* Definition part */

2007-03-07T04:36:56Z

Definition part

← Older revision		Revision as of 04:36, 7 March 2007
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	<math><R(X,Y)Y,X></math>		<math><R(X,Y)Y,X></math>

	viz the inner product of <math>R(X,Y)Y</math> and <math>X</math> with respect to <math>g</math>~~. This defines the sectional curvature~~.		viz the inner product of <math>R(X,Y)Y</math> and <math>X</math> with respect to <math>g</math>.

	''This ~~doesn't sound right~~. ~~Check it''~~		Divide this by the square of the area of the parallelogram formed by <math>X</math> and <math>Y</math>. This ratio defines the sectional curvature of <math>\Pi</math>, denoted as <math>K(\Pi)</math>.

	==Related notions==		==Related notions==