Totally real submanifold

Totally real submanifolds occur in complex geometry , a branch of mathematics . They generalize the concept of taking real vector space as a subspace of complex space . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

definition

It is an almost complex manifold . That is, is a smooth mapping of the tangent of on itself such that the restrictions for all , Vektorraumautomorphismen and are sufficient. ${\displaystyle (M,J)}$ ${\displaystyle J\colon TM\to TM}$ ${\displaystyle M}$ ${\displaystyle J|_{T_{x}M}\colon T_{x}M\to T_{x}M}$ ${\displaystyle x\in M}$ ${\displaystyle J|_{T_{x}M}\circ J|_{T_{x}M}=-\operatorname {id} _{T_{x}M}}$

An immersed submanifold of is now called ${\displaystyle N}$ ${\displaystyle M}$ totally real if holds for all . ${\displaystyle T_{y}N\cap J(T_{y}N)={\vec {0}}}$ ${\displaystyle y\in N}$

Of all the vectors that lie tangentially at the point , the almost complex structure therefore only maps the zero vector back onto a tangential vector of . Clearly speaking, the points of have only "real" tangential vectors and no actually "complex" ones . ${\displaystyle y}$ ${\displaystyle N}$ ${\displaystyle J}$ ${\displaystyle N}$ ${\displaystyle N}$

Examples

• ${\displaystyle \mathbb {R} ^{n}\subset \mathbb {C} ^{n}}$ is totally real.
• A Lagrangian submanifold of a symplectic manifold is totally real.

literature

• Bang-Yen Chen, "Riemannian submanifolds", 187-418. In: Handbook of differential geometry. Vol. I. Edited by Franki JE Dillen and Leopold CA Verstraelen. North-Holland, Amsterdam, 2000. ISBN 0-444-82240- 2
• Michèle Audin, François Lalonde, Leonid Polterovich: "Symplectic rigidity: Lagrangian submanifolds", 271-321. In: Holomorphic curves in symplectic geometry. Edited by Michèle Audin and Jacques Lafontaine. Progress in Mathematics, 117. Birkhäuser Verlag, Basel, 1994. ISBN 3-7643-2997-1